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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed d ≥ 1, is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for H-subgraph-free graphs where H is any finite set of graphs.

Carl Feghali, Felicia Lucke, Daniël Paulusma, and Bernard Ries. Matching Cuts in Graphs of High Girth and H-Free Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 31:1-31:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{feghali_et_al:LIPIcs.ISAAC.2023.31, author = {Feghali, Carl and Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Matching Cuts in Graphs of High Girth and H-Free Graphs}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.31}, URN = {urn:nbn:de:0030-drops-193332}, doi = {10.4230/LIPIcs.ISAAC.2023.31}, annote = {Keywords: matching cut, perfect matching, girth, H-free graph} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

For any finite set ℋ = {H_1,…,H_p} of graphs, a graph is ℋ-subgraph-free if it does not contain any of H_1,…,H_p as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of ℋ-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems.
We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 57:1-57:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{johnson_et_al:LIPIcs.MFCS.2023.57, author = {Johnson, Matthew and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Smith, Siani and van Leeuwen, Erik Jan}, title = {{Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.57}, URN = {urn:nbn:de:0030-drops-185914}, doi = {10.4230/LIPIcs.MFCS.2023.57}, annote = {Keywords: forbidden subgraphs, independent feedback vertex set, treewidth} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete, leading to many complexity results for both problems on special graph classes. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P₆+sP₂)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P₃-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs.

Felicia Lucke, Daniël Paulusma, and Bernard Ries. Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 64:1-64:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lucke_et_al:LIPIcs.MFCS.2023.64, author = {Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {64:1--64:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.64}, URN = {urn:nbn:de:0030-drops-185981}, doi = {10.4230/LIPIcs.MFCS.2023.64}, annote = {Keywords: matching cut, perfect matching, H-free graph, diameter, radius, dichotomy} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 11 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 22481 "Vertex Partitioning in Graphs: From Structure to Algorithms", which was held from 27 November to 2 December 2023. The report contains abstracts for presentations about recent structural and algorithmic developments for a variety of vertex partitioning problems. It also contains a collection of open problems which were posed during the seminar.

Maria Chudnovsky, Neeldhara Misra, Daniel Paulusma, Oliver Schaudt, and Akanksha Agrawal. Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481). In Dagstuhl Reports, Volume 12, Issue 11, pp. 109-123, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@Article{chudnovsky_et_al:DagRep.12.11.109, author = {Chudnovsky, Maria and Misra, Neeldhara and Paulusma, Daniel and Schaudt, Oliver and Agrawal, Akanksha}, title = {{Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481)}}, pages = {109--123}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {12}, number = {11}, editor = {Chudnovsky, Maria and Misra, Neeldhara and Paulusma, Daniel and Schaudt, Oliver and Agrawal, Akanksha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.11.109}, URN = {urn:nbn:de:0030-drops-178384}, doi = {10.4230/DagRep.12.11.109}, annote = {Keywords: computational complexity, hereditary graph classes, parameterized algorithms, polynomial-time algorithms, vertex partitioning} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r > 0 such that the first variant is NP-complete for P_r-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs.

Felicia Lucke, Daniël Paulusma, and Bernard Ries. Finding Matching Cuts in H-Free Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 22:1-22:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lucke_et_al:LIPIcs.ISAAC.2022.22, author = {Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Finding Matching Cuts in H-Free Graphs}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.22}, URN = {urn:nbn:de:0030-drops-173076}, doi = {10.4230/LIPIcs.ISAAC.2022.22}, annote = {Keywords: matching cut, perfect matching, H-free graph, computational complexity} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith. Partitioning H-Free Graphs of Bounded Diameter. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brause_et_al:LIPIcs.ISAAC.2021.21, author = {Brause, Christoph and Golovach, Petr and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Partitioning H-Free Graphs of Bounded Diameter}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.21}, URN = {urn:nbn:de:0030-drops-154543}, doi = {10.4230/LIPIcs.ISAAC.2021.21}, annote = {Keywords: vertex partitioning problem, H-free, diameter, complexity dichotomy} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{larose_et_al:LIPIcs.ESA.2021.58, author = {Larose, Beno\^{i}t and Markovi\'{c}, Petar and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani and \v{Z}ivn\'{y}, Stanislav}, title = {{QCSP on Reflexive Tournaments}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {58:1--58:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.58}, URN = {urn:nbn:de:0030-drops-146392}, doi = {10.4230/LIPIcs.ESA.2021.58}, annote = {Keywords: computational complexity, algorithmic graph theory, quantified constraints, universal algebra, constraint satisfaction} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for sP₃-free graphs for every integer s ≥ 1; here, the graph sP₃ denotes the disjoint union of s paths on three vertices. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs.

Giacomo Paesani, Daniël Paulusma, and Paweł Rzążewski. Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 82:1-82:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{paesani_et_al:LIPIcs.MFCS.2021.82, author = {Paesani, Giacomo and Paulusma, Dani\"{e}l and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {82:1--82:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.82}, URN = {urn:nbn:de:0030-drops-145224}, doi = {10.4230/LIPIcs.MFCS.2021.82}, annote = {Keywords: Feedback vertex set, even cycle transversal, odd cactus, forest, block} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration.
We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied.
We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H₁,H₂)-free graphs. We identify several general classes of (H₁,H₂)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H₁,H₂)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H₁ and H₂, the class of (H₁,H₂)-free graphs has unbounded mim-width.
Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H₁,H₂)-free graphs. In particular, we classify the mim-width of (H₁,H₂)-free graphs for all pairs (H₁,H₂) with |V(H₁)| + |V(H₂)| ≤ 8. When H₁ and H₂ are connected graphs, we classify all pairs (H₁,H₂) except for one remaining infinite family and a few isolated cases.

Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, and Daniël Paulusma. Bounding the Mim-Width of Hereditary Graph Classes. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 6:1-6:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brettell_et_al:LIPIcs.IPEC.2020.6, author = {Brettell, Nick and Horsfield, Jake and Munaro, Andrea and Paesani, Giacomo and Paulusma, Dani\"{e}l}, title = {{Bounding the Mim-Width of Hereditary Graph Classes}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.6}, URN = {urn:nbn:de:0030-drops-133099}, doi = {10.4230/LIPIcs.IPEC.2020.6}, annote = {Keywords: Width parameter, mim-width, clique-width, hereditary graph class} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction.

Walter Kern and Daniël Paulusma. Contracting to a Longest Path in H-Free Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 22:1-22:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kern_et_al:LIPIcs.ISAAC.2020.22, author = {Kern, Walter and Paulusma, Dani\"{e}l}, title = {{Contracting to a Longest Path in H-Free Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.22}, URN = {urn:nbn:de:0030-drops-133664}, doi = {10.4230/LIPIcs.ISAAC.2020.22}, annote = {Keywords: dichotomy, edge contraction, path, cycle, H-free graph} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

A k-colouring c of a graph G is a mapping V(G) → {1,2,… k} such that c(u) ≠ c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling).
A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years.
We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results:
1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs.
2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

Jan Bok, Nikola Jedlic̆ková, Barnaby Martin, Daniël Paulusma, and Siani Smith. Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 22:1-22:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bok_et_al:LIPIcs.ESA.2020.22, author = {Bok, Jan and Jedlic̆kov\'{a}, Nikola and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {22:1--22:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.22}, URN = {urn:nbn:de:0030-drops-128885}, doi = {10.4230/LIPIcs.ESA.2020.22}, annote = {Keywords: acyclic colouring, star colouring, injective colouring, H-free, dichotomy} }

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**Published in:** Dagstuhl Reports, Volume 9, Issue 6 (2020)

This report documents the program and the outcomes of Dagstuhl Seminar 19271 "Graph Colouring: from Structure to Algorithm", which was held from 30 June to 5 July 2019. The report contains abstracts for presentations about recent structural and algorithmic developments for the Graph Colouring problem and variants of it. It also contains a collection of open problems on graph colouring which were posed during the seminar.

Maria Chudnovsky, Daniel Paulusma, and Oliver Schaudt. Graph Colouring: from Structure to Algorithms (Dagstuhl Seminar 19271). In Dagstuhl Reports, Volume 9, Issue 6, pp. 125-142, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@Article{chudnovsky_et_al:DagRep.9.6.125, author = {Chudnovsky, Maria and Paulusma, Daniel and Schaudt, Oliver}, title = {{Graph Colouring: from Structure to Algorithms (Dagstuhl Seminar 19271)}}, pages = {125--142}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {6}, editor = {Chudnovsky, Maria and Paulusma, Daniel and Schaudt, Oliver}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.6.125}, URN = {urn:nbn:de:0030-drops-114905}, doi = {10.4230/DagRep.9.6.125}, annote = {Keywords: (certifying / parameterized / polynomial-time) algorithms, computational complexity, graph colouring, hereditary graph classes} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring H-Free Graphs of Bounded Diameter. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 14:1-14:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{martin_et_al:LIPIcs.MFCS.2019.14, author = {Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Colouring H-Free Graphs of Bounded Diameter}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.14}, URN = {urn:nbn:de:0030-drops-109584}, doi = {10.4230/LIPIcs.MFCS.2019.14}, annote = {Keywords: vertex colouring, H-free graph, diameter} }

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**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

A partition (V_1,...,V_k) of the vertex set of a graph G with a (not necessarily proper) colouring c is colourful if no two vertices in any V_i have the same colour and every set V_i induces a connected graph. The Colourful Partition problem, introduced by Adamaszek and Popa, is to decide whether a coloured graph (G,c) has a colourful partition of size at most k. This problem is related to the Colourful Components problem, introduced by He, Liu and Zhao, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most p edges.
Despite the similarities in their definitions, we show that Colourful Partition and Colourful Components may have different complexities for restricted instances. We tighten known NP-hardness results for both problems by closing a number of complexity gaps. In addition, we prove new hardness and tractability results for Colourful Partition. In particular, we prove that deciding whether a coloured graph (G,c) has a colourful partition of size 2 is NP-complete for coloured planar bipartite graphs of maximum degree 3 and path-width 3, but polynomial-time solvable for coloured graphs of treewidth 2.
Rather than performing an ad hoc study, we use our classical complexity results to guide us in undertaking a thorough parameterized study of Colourful Partition. We show that this leads to suitable parameters for obtaining FPT results and moreover prove that Colourful Components and Colourful Partition may have different parameterized complexities, depending on the chosen parameter.

Laurent Bulteau, Konrad K. Dabrowski, Guillaume Fertin, Matthew Johnson, Daniël Paulusma, and Stéphane Vialette. Finding a Small Number of Colourful Components. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 20:1-20:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2019.20, author = {Bulteau, Laurent and Dabrowski, Konrad K. and Fertin, Guillaume and Johnson, Matthew and Paulusma, Dani\"{e}l and Vialette, St\'{e}phane}, title = {{Finding a Small Number of Colourful Components}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.20}, URN = {urn:nbn:de:0030-drops-104914}, doi = {10.4230/LIPIcs.CPM.2019.20}, annote = {Keywords: Colourful component, colourful partition, tree, treewidth, vertex cover} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs.

Tereza Klimosová, Josef Malík, Tomás Masarík, Jana Novotná, Daniël Paulusma, and Veronika Slívová. Colouring (P_r+P_s)-Free Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 5:1-5:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{klimosova_et_al:LIPIcs.ISAAC.2018.5, author = {Klimosov\'{a}, Tereza and Mal{\'\i}k, Josef and Masar{\'\i}k, Tom\'{a}s and Novotn\'{a}, Jana and Paulusma, Dani\"{e}l and Sl{\'\i}vov\'{a}, Veronika}, title = {{Colouring (P\underliner+P\underlines)-Free Graphs}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {5:1--5:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.5}, URN = {urn:nbn:de:0030-drops-99533}, doi = {10.4230/LIPIcs.ISAAC.2018.5}, annote = {Keywords: vertex colouring, H-free graph, linear forest} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Let vc(G), fvs(G) and oct(G) denote, respectively, the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let, respectively, ivc(G), ifvs(G) or ioct(G) denote the minimum size of such a set. We investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases).

Konrad K. Dabrowski, Matthew Johnson, Giacomo Paesani, Daniël Paulusma, and Viktor Zamaraev. On the Price of Independence for Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 63:1-63:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dabrowski_et_al:LIPIcs.MFCS.2018.63, author = {Dabrowski, Konrad K. and Johnson, Matthew and Paesani, Giacomo and Paulusma, Dani\"{e}l and Zamaraev, Viktor}, title = {{On the Price of Independence for Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {63:1--63:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.63}, URN = {urn:nbn:de:0030-drops-96452}, doi = {10.4230/LIPIcs.MFCS.2018.63}, annote = {Keywords: vertex cover, feedback vertex set, odd cycle transversal, price of independence} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C_4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Barnaby Martin, Daniël Paulusma, and Erik Jan van Leeuwen. Disconnected Cuts in Claw-free Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 61:1-61:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{martin_et_al:LIPIcs.ESA.2018.61, author = {Martin, Barnaby and Paulusma, Dani\"{e}l and van Leeuwen, Erik Jan}, title = {{Disconnected Cuts in Claw-free Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {61:1--61:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.61}, URN = {urn:nbn:de:0030-drops-95249}, doi = {10.4230/LIPIcs.ESA.2018.61}, annote = {Keywords: disconnected cut, surjective homomorphism, biclique cover, claw-freeness} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a given integer k such that no two adjacent vertices are coloured alike. The complexity of Colouring is fully understood for graph classes characterized by one forbidden induced subgraph H. Despite a huge body of existing work, there are still major complexity gaps if two induced subgraphs H_1 and H_2 are forbidden. We let H_1 be the s-vertex cycle C_s and H_2 be the t-vertex path P_t. We show that Colouring is polynomial-time solvable for s=4 and t<=6, which unifies several known results for Colouring on (H_1,H_2)-free graphs. Our algorithm is based on a novel decomposition theorem for (C_4,P_6)-free graphs without clique cutsets into homogeneous pairs of sets and a new framework for bounding the clique-width of a graph by the clique-width of its subgraphs induced by homogeneous pairs of sets. To apply this framework, we also need to use divide-and-conquer to bound the clique-width of subgraphs induced by homogeneous pairs of sets. To complement our positive result we also prove that Colouring is NP-complete for s=4 and t>=9, which is the first hardness result on Colouring for (C_4,P_t)-free graphs.

Serge Gaspers, Shenwei Huang, and Daniel Paulusma. Colouring Square-Free Graphs without Long Induced Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 35:1-35:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gaspers_et_al:LIPIcs.STACS.2018.35, author = {Gaspers, Serge and Huang, Shenwei and Paulusma, Daniel}, title = {{Colouring Square-Free Graphs without Long Induced Paths}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {35:1--35:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.35}, URN = {urn:nbn:de:0030-drops-84922}, doi = {10.4230/LIPIcs.STACS.2018.35}, annote = {Keywords: graph colouring, hereditary graph class, clique-width, cycle, path} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs.
Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.
By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

Benoit Larose, Barnaby Martin, and Daniel Paulusma. Surjective H-Colouring over Reflexive Digraphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 49:1-49:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{larose_et_al:LIPIcs.STACS.2018.49, author = {Larose, Benoit and Martin, Barnaby and Paulusma, Daniel}, title = {{Surjective H-Colouring over Reflexive Digraphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.49}, URN = {urn:nbn:de:0030-drops-84882}, doi = {10.4230/LIPIcs.STACS.2018.49}, annote = {Keywords: Surjective H-Coloring, Computational Complexity, Algorithmic Graph Theory, Universal Algebra, Constraint Satisfaction} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

The NP-complete problem Feedback Vertex Set is to decide if it is possible, for a given integer k>=0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding if a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains in P for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks if a graph has an independent odd cycle transversal of size at most k for a given integer k>=0.

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent Feedback Vertex Set for P_5-free Graphs. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 16:1-16:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.ISAAC.2017.16, author = {Bonamy, Marthe and Dabrowski, Konrad K. and Feghali, Carl and Johnson, Matthew and Paulusma, Dani\"{e}l}, title = {{Independent Feedback Vertex Set for P\underline5-free Graphs}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {16:1--16:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.16}, URN = {urn:nbn:de:0030-drops-82308}, doi = {10.4230/LIPIcs.ISAAC.2017.16}, annote = {Keywords: feedback vertex set, odd cycle transversal, independent set, H-free graph} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Recognizing Graphs Close to Bipartite Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 70:1-70:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.MFCS.2017.70, author = {Bonamy, Marthe and Dabrowski, Konrad K. and Feghali, Carl and Johnson, Matthew and Paulusma, Dani\"{e}l}, title = {{Recognizing Graphs Close to Bipartite Graphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {70:1--70:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.70}, URN = {urn:nbn:de:0030-drops-80740}, doi = {10.4230/LIPIcs.MFCS.2017.70}, annote = {Keywords: degenerate graphs, near-bipartite graphs, reconfiguration graphs} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded).
Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2).
Finally, we discuss the consequences of our results for COLOURING.

Alexandre Blanché, Konrad K. Dabrowski, Matthew Johnson, Vadim V. Lozin, Daniël Paulusma, and Viktor Zamaraev. Clique-Width for Graph Classes Closed under Complementation. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 73:1-73:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blanche_et_al:LIPIcs.MFCS.2017.73, author = {Blanch\'{e}, Alexandre and Dabrowski, Konrad K. and Johnson, Matthew and Lozin, Vadim V. and Paulusma, Dani\"{e}l and Zamaraev, Viktor}, title = {{Clique-Width for Graph Classes Closed under Complementation}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {73:1--73:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.73}, URN = {urn:nbn:de:0030-drops-80756}, doi = {10.4230/LIPIcs.MFCS.2017.73}, annote = {Keywords: clique-width, self-complementary graph, forbidden induced subgraph} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs.

Paul Bonsma and Daniël Paulusma. Using Contracted Solution Graphs for Solving Reconfiguration Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 20:1-20:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bonsma_et_al:LIPIcs.MFCS.2016.20, author = {Bonsma, Paul and Paulusma, Dani\"{e}l}, title = {{Using Contracted Solution Graphs for Solving Reconfiguration Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.20}, URN = {urn:nbn:de:0030-drops-64351}, doi = {10.4230/LIPIcs.MFCS.2016.20}, annote = {Keywords: reconfiguration, contraction, dynamic programming, graph coloring} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph is a graph that can be made planar by the removal of at most k vertices. We prove that the generalization of Square Root, in which we are given two subsets of edges prescribed to be in or out of a square root, respectively, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, and Anthony Stewart. A Linear Kernel for Finding Square Roots of Almost Planar Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 4:1-4:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{golovach_et_al:LIPIcs.SWAT.2016.4, author = {Golovach, Petr A. and Kratsch, Dieter and Paulusma, Dani\"{e}l and Stewart, Anthony}, title = {{A Linear Kernel for Finding Square Roots of Almost Planar Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.4}, URN = {urn:nbn:de:0030-drops-60333}, doi = {10.4230/LIPIcs.SWAT.2016.4}, annote = {Keywords: planar graphs, square roots, linear kernel} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring Diamond-free Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dabrowski_et_al:LIPIcs.SWAT.2016.16, author = {Dabrowski, Konrad K. and Dross, Fran\c{c}ois and Paulusma, Dani\"{e}l}, title = {{Colouring Diamond-free Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.16}, URN = {urn:nbn:de:0030-drops-60380}, doi = {10.4230/LIPIcs.SWAT.2016.16}, annote = {Keywords: colouring, clique-width, diamond-free, graph class, hereditary graph class} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0.
Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value.
As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.

Konrad K. Dabrowski, Petr A. Golovach, Pim van 't Hof, and Daniel Paulusma. Editing to Eulerian Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 97-108, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dabrowski_et_al:LIPIcs.FSTTCS.2014.97, author = {Dabrowski, Konrad K. and Golovach, Petr A. and van 't Hof, Pim and Paulusma, Daniel}, title = {{Editing to Eulerian Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {97--108}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.97}, URN = {urn:nbn:de:0030-drops-48356}, doi = {10.4230/LIPIcs.FSTTCS.2014.97}, annote = {Keywords: Eulerian graphs, graph editing, polynomial algorithm} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity.

Daniel Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model Counting for CNF Formulas of Bounded Modular Treewidth. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 55-66, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{paulusma_et_al:LIPIcs.STACS.2013.55, author = {Paulusma, Daniel and Slivovsky, Friedrich and Szeider, Stefan}, title = {{Model Counting for CNF Formulas of Bounded Modular Treewidth}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {55--66}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.55}, URN = {urn:nbn:de:0030-drops-39226}, doi = {10.4230/LIPIcs.STACS.2013.55}, annote = {Keywords: Satisfiability, Model Counting, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with so-called ``beta-acyclic hypergraphs'' can be decided in polynomial time.
We also study the parameterized complexity of SAT for ``almost'' beta-acyclic instances, using as parameter the formula's distance from being beta-acyclic. As distance we use the size of smallest strong backdoor sets and the beta-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve (Discr. Appl. Math. 156, 2008).

Sebastian Ordyniak, Daniel Paulusma, and Stefan Szeider. Satisfiability of Acyclic and Almost Acyclic CNF Formulas. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 84-95, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{ordyniak_et_al:LIPIcs.FSTTCS.2010.84, author = {Ordyniak, Sebastian and Paulusma, Daniel and Szeider, Stefan}, title = {{Satisfiability of Acyclic and Almost Acyclic CNF Formulas}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {84--95}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.84}, URN = {urn:nbn:de:0030-drops-28556}, doi = {10.4230/LIPIcs.FSTTCS.2010.84}, annote = {Keywords: Satisfiability, chordal bipartite graphs, beta-acyclic hypergraphs, backdoor sets, parameterized complexity} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Testing whether there is an induced path in a graph spanning $k$ given vertices is already \NP-complete in general graphs when $k=3$. We show how to solve this problem in polynomial time on claw-free graphs, when $k$ is not part of the input but an arbitrarily fixed integer.

Jiri Fiala, Marcin Kaminski, Bernard Lidický, and Daniël Paulusma. The k-in-a-path Problem for Claw-free Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 371-382, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{fiala_et_al:LIPIcs.STACS.2010.2469, author = {Fiala, Jiri and Kaminski, Marcin and Lidick\'{y}, Bernard and Paulusma, Dani\"{e}l}, title = {{The k-in-a-path Problem for Claw-free Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {371--382}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2469}, URN = {urn:nbn:de:0030-drops-24692}, doi = {10.4230/LIPIcs.STACS.2010.2469}, annote = {Keywords: Induced path, claw-free graph, polynomial-time algorithm} }

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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed d ≥ 1, is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for H-subgraph-free graphs where H is any finite set of graphs.

Carl Feghali, Felicia Lucke, Daniël Paulusma, and Bernard Ries. Matching Cuts in Graphs of High Girth and H-Free Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 31:1-31:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{feghali_et_al:LIPIcs.ISAAC.2023.31, author = {Feghali, Carl and Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Matching Cuts in Graphs of High Girth and H-Free Graphs}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.31}, URN = {urn:nbn:de:0030-drops-193332}, doi = {10.4230/LIPIcs.ISAAC.2023.31}, annote = {Keywords: matching cut, perfect matching, girth, H-free graph} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

For any finite set ℋ = {H_1,…,H_p} of graphs, a graph is ℋ-subgraph-free if it does not contain any of H_1,…,H_p as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of ℋ-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems.
We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 57:1-57:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{johnson_et_al:LIPIcs.MFCS.2023.57, author = {Johnson, Matthew and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Smith, Siani and van Leeuwen, Erik Jan}, title = {{Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.57}, URN = {urn:nbn:de:0030-drops-185914}, doi = {10.4230/LIPIcs.MFCS.2023.57}, annote = {Keywords: forbidden subgraphs, independent feedback vertex set, treewidth} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete, leading to many complexity results for both problems on special graph classes. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P₆+sP₂)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P₃-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs.

Felicia Lucke, Daniël Paulusma, and Bernard Ries. Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 64:1-64:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lucke_et_al:LIPIcs.MFCS.2023.64, author = {Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {64:1--64:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.64}, URN = {urn:nbn:de:0030-drops-185981}, doi = {10.4230/LIPIcs.MFCS.2023.64}, annote = {Keywords: matching cut, perfect matching, H-free graph, diameter, radius, dichotomy} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 11 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 22481 "Vertex Partitioning in Graphs: From Structure to Algorithms", which was held from 27 November to 2 December 2023. The report contains abstracts for presentations about recent structural and algorithmic developments for a variety of vertex partitioning problems. It also contains a collection of open problems which were posed during the seminar.

Maria Chudnovsky, Neeldhara Misra, Daniel Paulusma, Oliver Schaudt, and Akanksha Agrawal. Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481). In Dagstuhl Reports, Volume 12, Issue 11, pp. 109-123, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@Article{chudnovsky_et_al:DagRep.12.11.109, author = {Chudnovsky, Maria and Misra, Neeldhara and Paulusma, Daniel and Schaudt, Oliver and Agrawal, Akanksha}, title = {{Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481)}}, pages = {109--123}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {12}, number = {11}, editor = {Chudnovsky, Maria and Misra, Neeldhara and Paulusma, Daniel and Schaudt, Oliver and Agrawal, Akanksha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.11.109}, URN = {urn:nbn:de:0030-drops-178384}, doi = {10.4230/DagRep.12.11.109}, annote = {Keywords: computational complexity, hereditary graph classes, parameterized algorithms, polynomial-time algorithms, vertex partitioning} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r > 0 such that the first variant is NP-complete for P_r-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs.

Felicia Lucke, Daniël Paulusma, and Bernard Ries. Finding Matching Cuts in H-Free Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 22:1-22:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lucke_et_al:LIPIcs.ISAAC.2022.22, author = {Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Finding Matching Cuts in H-Free Graphs}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.22}, URN = {urn:nbn:de:0030-drops-173076}, doi = {10.4230/LIPIcs.ISAAC.2022.22}, annote = {Keywords: matching cut, perfect matching, H-free graph, computational complexity} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith. Partitioning H-Free Graphs of Bounded Diameter. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brause_et_al:LIPIcs.ISAAC.2021.21, author = {Brause, Christoph and Golovach, Petr and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Partitioning H-Free Graphs of Bounded Diameter}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.21}, URN = {urn:nbn:de:0030-drops-154543}, doi = {10.4230/LIPIcs.ISAAC.2021.21}, annote = {Keywords: vertex partitioning problem, H-free, diameter, complexity dichotomy} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{larose_et_al:LIPIcs.ESA.2021.58, author = {Larose, Beno\^{i}t and Markovi\'{c}, Petar and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani and \v{Z}ivn\'{y}, Stanislav}, title = {{QCSP on Reflexive Tournaments}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {58:1--58:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.58}, URN = {urn:nbn:de:0030-drops-146392}, doi = {10.4230/LIPIcs.ESA.2021.58}, annote = {Keywords: computational complexity, algorithmic graph theory, quantified constraints, universal algebra, constraint satisfaction} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for sP₃-free graphs for every integer s ≥ 1; here, the graph sP₃ denotes the disjoint union of s paths on three vertices. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs.

Giacomo Paesani, Daniël Paulusma, and Paweł Rzążewski. Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 82:1-82:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{paesani_et_al:LIPIcs.MFCS.2021.82, author = {Paesani, Giacomo and Paulusma, Dani\"{e}l and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {82:1--82:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.82}, URN = {urn:nbn:de:0030-drops-145224}, doi = {10.4230/LIPIcs.MFCS.2021.82}, annote = {Keywords: Feedback vertex set, even cycle transversal, odd cactus, forest, block} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration.
We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied.
We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H₁,H₂)-free graphs. We identify several general classes of (H₁,H₂)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H₁,H₂)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H₁ and H₂, the class of (H₁,H₂)-free graphs has unbounded mim-width.
Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H₁,H₂)-free graphs. In particular, we classify the mim-width of (H₁,H₂)-free graphs for all pairs (H₁,H₂) with |V(H₁)| + |V(H₂)| ≤ 8. When H₁ and H₂ are connected graphs, we classify all pairs (H₁,H₂) except for one remaining infinite family and a few isolated cases.

Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, and Daniël Paulusma. Bounding the Mim-Width of Hereditary Graph Classes. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 6:1-6:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brettell_et_al:LIPIcs.IPEC.2020.6, author = {Brettell, Nick and Horsfield, Jake and Munaro, Andrea and Paesani, Giacomo and Paulusma, Dani\"{e}l}, title = {{Bounding the Mim-Width of Hereditary Graph Classes}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.6}, URN = {urn:nbn:de:0030-drops-133099}, doi = {10.4230/LIPIcs.IPEC.2020.6}, annote = {Keywords: Width parameter, mim-width, clique-width, hereditary graph class} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction.

Walter Kern and Daniël Paulusma. Contracting to a Longest Path in H-Free Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 22:1-22:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kern_et_al:LIPIcs.ISAAC.2020.22, author = {Kern, Walter and Paulusma, Dani\"{e}l}, title = {{Contracting to a Longest Path in H-Free Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.22}, URN = {urn:nbn:de:0030-drops-133664}, doi = {10.4230/LIPIcs.ISAAC.2020.22}, annote = {Keywords: dichotomy, edge contraction, path, cycle, H-free graph} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

A k-colouring c of a graph G is a mapping V(G) → {1,2,… k} such that c(u) ≠ c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling).
A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years.
We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results:
1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs.
2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

Jan Bok, Nikola Jedlic̆ková, Barnaby Martin, Daniël Paulusma, and Siani Smith. Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 22:1-22:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bok_et_al:LIPIcs.ESA.2020.22, author = {Bok, Jan and Jedlic̆kov\'{a}, Nikola and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {22:1--22:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.22}, URN = {urn:nbn:de:0030-drops-128885}, doi = {10.4230/LIPIcs.ESA.2020.22}, annote = {Keywords: acyclic colouring, star colouring, injective colouring, H-free, dichotomy} }

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**Published in:** Dagstuhl Reports, Volume 9, Issue 6 (2020)

This report documents the program and the outcomes of Dagstuhl Seminar 19271 "Graph Colouring: from Structure to Algorithm", which was held from 30 June to 5 July 2019. The report contains abstracts for presentations about recent structural and algorithmic developments for the Graph Colouring problem and variants of it. It also contains a collection of open problems on graph colouring which were posed during the seminar.

Maria Chudnovsky, Daniel Paulusma, and Oliver Schaudt. Graph Colouring: from Structure to Algorithms (Dagstuhl Seminar 19271). In Dagstuhl Reports, Volume 9, Issue 6, pp. 125-142, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@Article{chudnovsky_et_al:DagRep.9.6.125, author = {Chudnovsky, Maria and Paulusma, Daniel and Schaudt, Oliver}, title = {{Graph Colouring: from Structure to Algorithms (Dagstuhl Seminar 19271)}}, pages = {125--142}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {6}, editor = {Chudnovsky, Maria and Paulusma, Daniel and Schaudt, Oliver}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.6.125}, URN = {urn:nbn:de:0030-drops-114905}, doi = {10.4230/DagRep.9.6.125}, annote = {Keywords: (certifying / parameterized / polynomial-time) algorithms, computational complexity, graph colouring, hereditary graph classes} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring H-Free Graphs of Bounded Diameter. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 14:1-14:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{martin_et_al:LIPIcs.MFCS.2019.14, author = {Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Colouring H-Free Graphs of Bounded Diameter}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.14}, URN = {urn:nbn:de:0030-drops-109584}, doi = {10.4230/LIPIcs.MFCS.2019.14}, annote = {Keywords: vertex colouring, H-free graph, diameter} }

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**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

A partition (V_1,...,V_k) of the vertex set of a graph G with a (not necessarily proper) colouring c is colourful if no two vertices in any V_i have the same colour and every set V_i induces a connected graph. The Colourful Partition problem, introduced by Adamaszek and Popa, is to decide whether a coloured graph (G,c) has a colourful partition of size at most k. This problem is related to the Colourful Components problem, introduced by He, Liu and Zhao, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most p edges.
Despite the similarities in their definitions, we show that Colourful Partition and Colourful Components may have different complexities for restricted instances. We tighten known NP-hardness results for both problems by closing a number of complexity gaps. In addition, we prove new hardness and tractability results for Colourful Partition. In particular, we prove that deciding whether a coloured graph (G,c) has a colourful partition of size 2 is NP-complete for coloured planar bipartite graphs of maximum degree 3 and path-width 3, but polynomial-time solvable for coloured graphs of treewidth 2.
Rather than performing an ad hoc study, we use our classical complexity results to guide us in undertaking a thorough parameterized study of Colourful Partition. We show that this leads to suitable parameters for obtaining FPT results and moreover prove that Colourful Components and Colourful Partition may have different parameterized complexities, depending on the chosen parameter.

Laurent Bulteau, Konrad K. Dabrowski, Guillaume Fertin, Matthew Johnson, Daniël Paulusma, and Stéphane Vialette. Finding a Small Number of Colourful Components. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 20:1-20:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2019.20, author = {Bulteau, Laurent and Dabrowski, Konrad K. and Fertin, Guillaume and Johnson, Matthew and Paulusma, Dani\"{e}l and Vialette, St\'{e}phane}, title = {{Finding a Small Number of Colourful Components}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.20}, URN = {urn:nbn:de:0030-drops-104914}, doi = {10.4230/LIPIcs.CPM.2019.20}, annote = {Keywords: Colourful component, colourful partition, tree, treewidth, vertex cover} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs.

Tereza Klimosová, Josef Malík, Tomás Masarík, Jana Novotná, Daniël Paulusma, and Veronika Slívová. Colouring (P_r+P_s)-Free Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 5:1-5:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{klimosova_et_al:LIPIcs.ISAAC.2018.5, author = {Klimosov\'{a}, Tereza and Mal{\'\i}k, Josef and Masar{\'\i}k, Tom\'{a}s and Novotn\'{a}, Jana and Paulusma, Dani\"{e}l and Sl{\'\i}vov\'{a}, Veronika}, title = {{Colouring (P\underliner+P\underlines)-Free Graphs}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {5:1--5:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.5}, URN = {urn:nbn:de:0030-drops-99533}, doi = {10.4230/LIPIcs.ISAAC.2018.5}, annote = {Keywords: vertex colouring, H-free graph, linear forest} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Let vc(G), fvs(G) and oct(G) denote, respectively, the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let, respectively, ivc(G), ifvs(G) or ioct(G) denote the minimum size of such a set. We investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases).

Konrad K. Dabrowski, Matthew Johnson, Giacomo Paesani, Daniël Paulusma, and Viktor Zamaraev. On the Price of Independence for Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 63:1-63:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dabrowski_et_al:LIPIcs.MFCS.2018.63, author = {Dabrowski, Konrad K. and Johnson, Matthew and Paesani, Giacomo and Paulusma, Dani\"{e}l and Zamaraev, Viktor}, title = {{On the Price of Independence for Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {63:1--63:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.63}, URN = {urn:nbn:de:0030-drops-96452}, doi = {10.4230/LIPIcs.MFCS.2018.63}, annote = {Keywords: vertex cover, feedback vertex set, odd cycle transversal, price of independence} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C_4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Barnaby Martin, Daniël Paulusma, and Erik Jan van Leeuwen. Disconnected Cuts in Claw-free Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 61:1-61:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{martin_et_al:LIPIcs.ESA.2018.61, author = {Martin, Barnaby and Paulusma, Dani\"{e}l and van Leeuwen, Erik Jan}, title = {{Disconnected Cuts in Claw-free Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {61:1--61:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.61}, URN = {urn:nbn:de:0030-drops-95249}, doi = {10.4230/LIPIcs.ESA.2018.61}, annote = {Keywords: disconnected cut, surjective homomorphism, biclique cover, claw-freeness} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a given integer k such that no two adjacent vertices are coloured alike. The complexity of Colouring is fully understood for graph classes characterized by one forbidden induced subgraph H. Despite a huge body of existing work, there are still major complexity gaps if two induced subgraphs H_1 and H_2 are forbidden. We let H_1 be the s-vertex cycle C_s and H_2 be the t-vertex path P_t. We show that Colouring is polynomial-time solvable for s=4 and t<=6, which unifies several known results for Colouring on (H_1,H_2)-free graphs. Our algorithm is based on a novel decomposition theorem for (C_4,P_6)-free graphs without clique cutsets into homogeneous pairs of sets and a new framework for bounding the clique-width of a graph by the clique-width of its subgraphs induced by homogeneous pairs of sets. To apply this framework, we also need to use divide-and-conquer to bound the clique-width of subgraphs induced by homogeneous pairs of sets. To complement our positive result we also prove that Colouring is NP-complete for s=4 and t>=9, which is the first hardness result on Colouring for (C_4,P_t)-free graphs.

Serge Gaspers, Shenwei Huang, and Daniel Paulusma. Colouring Square-Free Graphs without Long Induced Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 35:1-35:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gaspers_et_al:LIPIcs.STACS.2018.35, author = {Gaspers, Serge and Huang, Shenwei and Paulusma, Daniel}, title = {{Colouring Square-Free Graphs without Long Induced Paths}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {35:1--35:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.35}, URN = {urn:nbn:de:0030-drops-84922}, doi = {10.4230/LIPIcs.STACS.2018.35}, annote = {Keywords: graph colouring, hereditary graph class, clique-width, cycle, path} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs.
Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.
By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

Benoit Larose, Barnaby Martin, and Daniel Paulusma. Surjective H-Colouring over Reflexive Digraphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 49:1-49:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{larose_et_al:LIPIcs.STACS.2018.49, author = {Larose, Benoit and Martin, Barnaby and Paulusma, Daniel}, title = {{Surjective H-Colouring over Reflexive Digraphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.49}, URN = {urn:nbn:de:0030-drops-84882}, doi = {10.4230/LIPIcs.STACS.2018.49}, annote = {Keywords: Surjective H-Coloring, Computational Complexity, Algorithmic Graph Theory, Universal Algebra, Constraint Satisfaction} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

The NP-complete problem Feedback Vertex Set is to decide if it is possible, for a given integer k>=0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding if a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains in P for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks if a graph has an independent odd cycle transversal of size at most k for a given integer k>=0.

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent Feedback Vertex Set for P_5-free Graphs. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 16:1-16:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.ISAAC.2017.16, author = {Bonamy, Marthe and Dabrowski, Konrad K. and Feghali, Carl and Johnson, Matthew and Paulusma, Dani\"{e}l}, title = {{Independent Feedback Vertex Set for P\underline5-free Graphs}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {16:1--16:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.16}, URN = {urn:nbn:de:0030-drops-82308}, doi = {10.4230/LIPIcs.ISAAC.2017.16}, annote = {Keywords: feedback vertex set, odd cycle transversal, independent set, H-free graph} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Recognizing Graphs Close to Bipartite Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 70:1-70:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.MFCS.2017.70, author = {Bonamy, Marthe and Dabrowski, Konrad K. and Feghali, Carl and Johnson, Matthew and Paulusma, Dani\"{e}l}, title = {{Recognizing Graphs Close to Bipartite Graphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {70:1--70:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.70}, URN = {urn:nbn:de:0030-drops-80740}, doi = {10.4230/LIPIcs.MFCS.2017.70}, annote = {Keywords: degenerate graphs, near-bipartite graphs, reconfiguration graphs} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded).
Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2).
Finally, we discuss the consequences of our results for COLOURING.

Alexandre Blanché, Konrad K. Dabrowski, Matthew Johnson, Vadim V. Lozin, Daniël Paulusma, and Viktor Zamaraev. Clique-Width for Graph Classes Closed under Complementation. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 73:1-73:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blanche_et_al:LIPIcs.MFCS.2017.73, author = {Blanch\'{e}, Alexandre and Dabrowski, Konrad K. and Johnson, Matthew and Lozin, Vadim V. and Paulusma, Dani\"{e}l and Zamaraev, Viktor}, title = {{Clique-Width for Graph Classes Closed under Complementation}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {73:1--73:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.73}, URN = {urn:nbn:de:0030-drops-80756}, doi = {10.4230/LIPIcs.MFCS.2017.73}, annote = {Keywords: clique-width, self-complementary graph, forbidden induced subgraph} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs.

Paul Bonsma and Daniël Paulusma. Using Contracted Solution Graphs for Solving Reconfiguration Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 20:1-20:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bonsma_et_al:LIPIcs.MFCS.2016.20, author = {Bonsma, Paul and Paulusma, Dani\"{e}l}, title = {{Using Contracted Solution Graphs for Solving Reconfiguration Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.20}, URN = {urn:nbn:de:0030-drops-64351}, doi = {10.4230/LIPIcs.MFCS.2016.20}, annote = {Keywords: reconfiguration, contraction, dynamic programming, graph coloring} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph is a graph that can be made planar by the removal of at most k vertices. We prove that the generalization of Square Root, in which we are given two subsets of edges prescribed to be in or out of a square root, respectively, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, and Anthony Stewart. A Linear Kernel for Finding Square Roots of Almost Planar Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 4:1-4:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{golovach_et_al:LIPIcs.SWAT.2016.4, author = {Golovach, Petr A. and Kratsch, Dieter and Paulusma, Dani\"{e}l and Stewart, Anthony}, title = {{A Linear Kernel for Finding Square Roots of Almost Planar Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.4}, URN = {urn:nbn:de:0030-drops-60333}, doi = {10.4230/LIPIcs.SWAT.2016.4}, annote = {Keywords: planar graphs, square roots, linear kernel} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring Diamond-free Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dabrowski_et_al:LIPIcs.SWAT.2016.16, author = {Dabrowski, Konrad K. and Dross, Fran\c{c}ois and Paulusma, Dani\"{e}l}, title = {{Colouring Diamond-free Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.16}, URN = {urn:nbn:de:0030-drops-60380}, doi = {10.4230/LIPIcs.SWAT.2016.16}, annote = {Keywords: colouring, clique-width, diamond-free, graph class, hereditary graph class} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0.
Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value.
As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.

Konrad K. Dabrowski, Petr A. Golovach, Pim van 't Hof, and Daniel Paulusma. Editing to Eulerian Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 97-108, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dabrowski_et_al:LIPIcs.FSTTCS.2014.97, author = {Dabrowski, Konrad K. and Golovach, Petr A. and van 't Hof, Pim and Paulusma, Daniel}, title = {{Editing to Eulerian Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {97--108}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.97}, URN = {urn:nbn:de:0030-drops-48356}, doi = {10.4230/LIPIcs.FSTTCS.2014.97}, annote = {Keywords: Eulerian graphs, graph editing, polynomial algorithm} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity.

Daniel Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model Counting for CNF Formulas of Bounded Modular Treewidth. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 55-66, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{paulusma_et_al:LIPIcs.STACS.2013.55, author = {Paulusma, Daniel and Slivovsky, Friedrich and Szeider, Stefan}, title = {{Model Counting for CNF Formulas of Bounded Modular Treewidth}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {55--66}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.55}, URN = {urn:nbn:de:0030-drops-39226}, doi = {10.4230/LIPIcs.STACS.2013.55}, annote = {Keywords: Satisfiability, Model Counting, Parameterized Complexity} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with so-called ``beta-acyclic hypergraphs'' can be decided in polynomial time.
We also study the parameterized complexity of SAT for ``almost'' beta-acyclic instances, using as parameter the formula's distance from being beta-acyclic. As distance we use the size of smallest strong backdoor sets and the beta-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve (Discr. Appl. Math. 156, 2008).

Sebastian Ordyniak, Daniel Paulusma, and Stefan Szeider. Satisfiability of Acyclic and Almost Acyclic CNF Formulas. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 84-95, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{ordyniak_et_al:LIPIcs.FSTTCS.2010.84, author = {Ordyniak, Sebastian and Paulusma, Daniel and Szeider, Stefan}, title = {{Satisfiability of Acyclic and Almost Acyclic CNF Formulas}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {84--95}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.84}, URN = {urn:nbn:de:0030-drops-28556}, doi = {10.4230/LIPIcs.FSTTCS.2010.84}, annote = {Keywords: Satisfiability, chordal bipartite graphs, beta-acyclic hypergraphs, backdoor sets, parameterized complexity} }

Document

**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Testing whether there is an induced path in a graph spanning $k$ given vertices is already \NP-complete in general graphs when $k=3$. We show how to solve this problem in polynomial time on claw-free graphs, when $k$ is not part of the input but an arbitrarily fixed integer.

Jiri Fiala, Marcin Kaminski, Bernard Lidický, and Daniël Paulusma. The k-in-a-path Problem for Claw-free Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 371-382, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{fiala_et_al:LIPIcs.STACS.2010.2469, author = {Fiala, Jiri and Kaminski, Marcin and Lidick\'{y}, Bernard and Paulusma, Dani\"{e}l}, title = {{The k-in-a-path Problem for Claw-free Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {371--382}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2469}, URN = {urn:nbn:de:0030-drops-24692}, doi = {10.4230/LIPIcs.STACS.2010.2469}, annote = {Keywords: Induced path, claw-free graph, polynomial-time algorithm} }

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