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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

A book embedding of a graph is a drawing that maps vertices onto a line and edges to simple pairwise non-crossing curves drawn into "pages", which are half-planes bounded by that line. Two-page book embeddings, i.e., book embeddings into 2 pages, are of special importance as they are both NP-hard to compute and have specific applications. We obtain a 2^𝒪(√n) algorithm for computing a book embedding of an n-vertex graph on two pages - a result which is asymptotically tight under the Exponential Time Hypothesis. As a key tool in our approach, we obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph. We conclude by establishing the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question arising from previous work on the problem.

Robert Ganian, Haiko Müller, Sebastian Ordyniak, Giacomo Paesani, and Mateusz Rychlicki. A Tight Subexponential-Time Algorithm for Two-Page Book Embedding. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 68:1-68:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ganian_et_al:LIPIcs.ICALP.2024.68, author = {Ganian, Robert and M\"{u}ller, Haiko and Ordyniak, Sebastian and Paesani, Giacomo and Rychlicki, Mateusz}, title = {{A Tight Subexponential-Time Algorithm for Two-Page Book Embedding}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {68:1--68:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.68}, URN = {urn:nbn:de:0030-drops-202114}, doi = {10.4230/LIPIcs.ICALP.2024.68}, annote = {Keywords: book embedding, page number, subexponential algorithms, subhamiltonicity, feedback edge number} }

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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 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} }