31 Search Results for "Paulusma, Daniël"

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.47
Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-Graphs

Authors: Vadim Lozin, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Mark Siggers, Siani Smith, and Erik Jan van Leeuwen

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
For a fixed set H of graphs, a graph G is H-subgraph-free if G does not contain any H ∈ H as a (not necessarily induced) subgraph. A recent framework gives a complete classification on H-subgraph-free graphs (for finite sets H) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity in H-subgraph-free graphs is unknown. We study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: Hamilton Cycle, k-Induced Disjoint Paths, C₅-Colouring and Star 3-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and also from problems that do satisfy all three conditions of the framework, in particular when we forbid certain subdivisions of the "H"-graph (the graph that looks like the letter "H"). Hence, we exhibit a rich complexity landscape among problems for H-subgraph-free graph classes.
Cite as

Vadim Lozin, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Mark Siggers, Siani Smith, and Erik Jan van Leeuwen. Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lozin_et_al:LIPIcs.ISAAC.2024.47,
  author =	{Lozin, Vadim and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Siggers, Mark and Smith, Siani and van Leeuwen, Erik Jan},
  title =	{{Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-Graphs}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{47:1--47:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.47},
  URN =		{urn:nbn:de:0030-drops-221747},
  doi =		{10.4230/LIPIcs.ISAAC.2024.47},
  annote =	{Keywords: forbidden subgraph, complexity dichotomy, edge subdivision, treewidth}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.29
Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs

Authors: Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
It is known that the weighted version of Edge Multiway Cut (also known as Multiterminal Cut) is NP-complete on planar graphs of maximum degree 3. In contrast, for the unweighted version, NP-completeness is only known for planar graphs of maximum degree 11. In fact, the complexity of unweighted Edge Multiway Cut was open for graphs of maximum degree 3 for over twenty years. We prove that the unweighted version is NP-complete even for planar graphs of maximum degree 3. As weighted Edge Multiway Cut is polynomial-time solvable for graphs of maximum degree at most 2, we have now closed the complexity gap. We also prove that (unweighted) Node Multiway Cut (both with and without deletable terminals) is NP-complete for planar graphs of maximum degree 3. By combining our results with known results, we can apply two meta-classifications on graph containment from the literature. This yields full dichotomies for all three problems on H-topological-minor-free graphs and, should H be finite, on H-subgraph-free graphs as well. Previously, such dichotomies were only implied for H-minor-free graphs.
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Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{johnson_et_al:LIPIcs.SWAT.2024.29,
  author =	{Johnson, Matthew and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Smith, Siani and van Leeuwen, Erik Jan},
  title =	{{Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.29},
  URN =		{urn:nbn:de:0030-drops-200699},
  doi =		{10.4230/LIPIcs.SWAT.2024.29},
  annote =	{Keywords: multiway cut, planar subcubic graph, complexity dichotomy, graph containment}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.31
Matching Cuts in Graphs of High Girth and H-Free Graphs

Authors: Carl Feghali, Felicia Lucke, Daniël Paulusma, and Bernard Ries

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

Abstract
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.
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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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.57
Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs

Authors: Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.64
Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius

Authors: Felicia Lucke, Daniël Paulusma, and Bernard Ries

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/DagRep.12.11.109
Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481)

Authors: Maria Chudnovsky, Neeldhara Misra, Daniel Paulusma, Oliver Schaudt, and Akanksha Agrawal

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.22
Finding Matching Cuts in H-Free Graphs

Authors: Felicia Lucke, Daniël Paulusma, and Bernard Ries

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.21
Partitioning H-Free Graphs of Bounded Diameter

Authors: Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.58
QCSP on Reflexive Tournaments

Authors: Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.82
Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

Authors: Giacomo Paesani, Daniël Paulusma, and Paweł Rzążewski

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

Abstract
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.
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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}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.6
Bounding the Mim-Width of Hereditary Graph Classes

Authors: Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, and Daniël Paulusma

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.22
Contracting to a Longest Path in H-Free Graphs

Authors: Walter Kern and Daniël Paulusma

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.22
Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs

Authors: Jan Bok, Nikola Jedlic̆ková, Barnaby Martin, Daniël Paulusma, and Siani Smith

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/DagRep.9.6.125
Graph Colouring: from Structure to Algorithms (Dagstuhl Seminar 19271)

Authors: Maria Chudnovsky, Daniel Paulusma, and Oliver Schaudt

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.14
Colouring H-Free Graphs of Bounded Diameter

Authors: Barnaby Martin, Daniël Paulusma, and Siani Smith

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

Abstract
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.
Cite as

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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