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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.

Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý. Treewidth Is NP-Complete on Cubic Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.7, author = {Bodlaender, Hans L. and Bonnet, \'{E}douard and Jaffke, Lars and Knop, Du\v{s}an and Lima, Paloma T. and Milani\v{c}, Martin and Ordyniak, Sebastian and Pandey, Sukanya and Such\'{y}, Ond\v{r}ej}, title = {{Treewidth Is NP-Complete on Cubic Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.7}, URN = {urn:nbn:de:0030-drops-194263}, doi = {10.4230/LIPIcs.IPEC.2023.7}, annote = {Keywords: Treewidth, cubic graphs, degree, NP-completeness} }

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

The b-Coloring problem, which given a graph G and an integer k asks whether G has a proper k-coloring such that each color class has a vertex adjacent to all color classes except its own, is known to be FPT parameterized by the vertex cover number and XP and 𝖶[1]-hard parameterized by clique-width. Its complexity when parameterized by the treewidth of the input graph remained an open problem. We settle this question by showing that b-Coloring is XNLP-complete when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is 𝖶[t]-hard for all t, and resolves the parameterized complexity of b-Coloring parameterized by treewidth. We complement this result by showing that b-Coloring is FPT when parameterized by neighborhood diversity and by twin cover, two parameters that generalize vertex cover to more dense graphs, but are incomparable to pathwidth.

Lars Jaffke, Paloma T. Lima, and Roohani Sharma. Structural Parameterizations of b-Coloring. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jaffke_et_al:LIPIcs.ISAAC.2023.40, author = {Jaffke, Lars and Lima, Paloma T. and Sharma, Roohani}, title = {{Structural Parameterizations of b-Coloring}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {40:1--40:14}, 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.2023.40}, URN = {urn:nbn:de:0030-drops-193429}, doi = {10.4230/LIPIcs.ISAAC.2023.40}, annote = {Keywords: b-coloring, structural parameterization, XNLP, pathwidth, neighborhood diversity, twin cover} }

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

Given a graph and a colouring of its vertices, a rainbow vertex path is a path between two vertices such that all the internal nodes of the path are coloured distinctly. A graph is rainbow vertex-connected if between every pair of vertices in the graph there exists a rainbow vertex path. We study the problem of deciding whether a given graph can be coloured using k or less colours such that it is rainbow vertex-connected. Note that every graph G needs at least diam(G)-1 colours to be rainbow vertex connected.
Heggernes et al. [MFCS, 2018] conjectured that if G is a graph in which every induced subgraph has a dominating diametral path, then G can always be rainbow vertex coloured with diam(G)-1 many colours. In this work, we confirm their conjecture for chordal, bipartite and claw-free diametral path graphs. We complement these results by showing the conjecture does not hold if the condition on every induced subgraph is dropped. In fact we show that, in this case, even though diam(G) many colours are always enough, it is NP-complete to determine whether a graph with a dominating diametral path of length three can be rainbow vertex coloured with two colours.

Jakob Dyrseth and Paloma T. Lima. On the Complexity of Rainbow Vertex Colouring Diametral Path Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dyrseth_et_al:LIPIcs.ISAAC.2022.43, author = {Dyrseth, Jakob and Lima, Paloma T.}, title = {{On the Complexity of Rainbow Vertex Colouring Diametral Path Graphs}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {43:1--43:13}, 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.2022.43}, URN = {urn:nbn:de:0030-drops-173286}, doi = {10.4230/LIPIcs.ISAAC.2022.43}, annote = {Keywords: rainbow vertex colouring, diametral path graphs, interval graphs} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space.
In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.

Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.8, author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo and Jaffke, Lars and Lima, Paloma T.}, title = {{XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.8}, URN = {urn:nbn:de:0030-drops-173640}, doi = {10.4230/LIPIcs.IPEC.2022.8}, annote = {Keywords: parameterized complexity, XNLP, linear clique-width, W-hierarchy, pathwidth, linear mim-width, bandwidth} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class 𝒢 there exists a constant k such that no member of 𝒢 contains a k-creature as an induced subgraph or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every G ∈ 𝒢 contains at most p(|V(G)|) minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from 𝒢. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

Jakub Gajarský, Lars Jaffke, Paloma T. Lima, Jana Novotná, Marcin Pilipczuk, Paweł Rzążewski, and Uéverton S. Souza. Taming Graphs with No Large Creatures and Skinny Ladders. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 58:1-58:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gajarsky_et_al:LIPIcs.ESA.2022.58, author = {Gajarsk\'{y}, Jakub and Jaffke, Lars and Lima, Paloma T. and Novotn\'{a}, Jana and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l} and Souza, U\'{e}verton S.}, title = {{Taming Graphs with No Large Creatures and Skinny Ladders}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {58:1--58:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.58}, URN = {urn:nbn:de:0030-drops-169969}, doi = {10.4230/LIPIcs.ESA.2022.58}, annote = {Keywords: Minimal separator, hereditary graph class} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The Contraction(vc) problem takes as input a graph G on n vertices and two integers k and d, and asks whether one can contract at most k edges to reduce the size of a minimum vertex cover of G by at least d. Recently, Lima et al. [MFCS 2020, JCSS 2021] proved, among other results, that unlike most of the so-called blocker problems, Contraction(vc) admits an XP algorithm running in time f(d) ⋅ n^O(d). They left open the question of whether this problem is FPT under this parameterization. In this article, we continue this line of research and prove the following results:
- Contraction(vc) is W[1]-hard parameterized by k + d. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time f(k + d) ⋅ n^o(k + d) for any function f. In particular, this answers the open question stated in Lima et al. [MFCS 2020] in the negative.
- It is NP-hard to decide whether an instance (G, k, d) of {Contraction(vc)} is a Yes-instance even when k = d, hence enhancing our understanding of the classical complexity of the problem.
- Contraction(vc) can be solved in time 2^O(d) ⋅ n^{k - d + O(1)}. This XP algorithm improves the one of Lima et al. [MFCS 2020], which uses Courcelle’s theorem as a subroutine and hence, the f(d)-factor in the running time is non-explicit and probably very large. On the other hand, this shows that when k = d, the problem is FPT parameterized by d (or by k).

Paloma T. Lima, Vinicius F. dos Santos, Ignasi Sau, Uéverton S. Souza, and Prafullkumar Tale. Reducing the Vertex Cover Number via Edge Contractions. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lima_et_al:LIPIcs.MFCS.2022.69, author = {Lima, Paloma T. and dos Santos, Vinicius F. and Sau, Ignasi and Souza, U\'{e}verton S. and Tale, Prafullkumar}, title = {{Reducing the Vertex Cover Number via Edge Contractions}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {69:1--69:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.69}, URN = {urn:nbn:de:0030-drops-168671}, doi = {10.4230/LIPIcs.MFCS.2022.69}, annote = {Keywords: Blocker problems, edge contraction, vertex cover, parameterized complexity} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial-time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for b-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.

Lars Jaffke, Paloma T. Lima, and Daniel Lokshtanov. b-Coloring Parameterized by Clique-Width. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jaffke_et_al:LIPIcs.STACS.2021.43, author = {Jaffke, Lars and Lima, Paloma T. and Lokshtanov, Daniel}, title = {{b-Coloring Parameterized by Clique-Width}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.43}, URN = {urn:nbn:de:0030-drops-136881}, doi = {10.4230/LIPIcs.STACS.2021.43}, annote = {Keywords: b-Coloring, clique-width, vertex cover, structural parameterization} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an 𝒪^⋆(q^{tw})-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width.

Lars Jaffke, Paloma T. Lima, and Geevarghese Philip. Structural Parameterizations of Clique Coloring. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2020.49, author = {Jaffke, Lars and Lima, Paloma T. and Philip, Geevarghese}, title = {{Structural Parameterizations of Clique Coloring}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {49:1--49:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.49}, URN = {urn:nbn:de:0030-drops-127157}, doi = {10.4230/LIPIcs.MFCS.2020.49}, annote = {Keywords: clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed p ≥ 3 both variants of the problem become NP-complete when restricted to split (S₃,…,S_p)-free graphs, where S_q denotes the q-sun graph.

Paloma T. Lima, Erik Jan van Leeuwen, and Marieke van der Wegen. Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lima_et_al:LIPIcs.MFCS.2020.63, author = {Lima, Paloma T. and van Leeuwen, Erik Jan and van der Wegen, Marieke}, title = {{Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {63:1--63:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.63}, URN = {urn:nbn:de:0030-drops-127331}, doi = {10.4230/LIPIcs.MFCS.2020.63}, annote = {Keywords: rainbow vertex coloring, permutation graphs, powers of trees} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

For a graph parameter π, the Contraction(π) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where π is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ℋ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ℋ, which in particular imply that Contraction(π) is co-NP-hard even for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Paloma T. Lima, Vinicius F. dos Santos, Ignasi Sau, and Uéverton S. Souza. Reducing Graph Transversals via Edge Contractions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 64:1-64:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lima_et_al:LIPIcs.MFCS.2020.64, author = {Lima, Paloma T. and dos Santos, Vinicius F. and Sau, Ignasi and Souza, U\'{e}verton S.}, title = {{Reducing Graph Transversals via Edge Contractions}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {64:1--64:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.64}, URN = {urn:nbn:de:0030-drops-127346}, doi = {10.4230/LIPIcs.MFCS.2020.64}, annote = {Keywords: blocker problem, edge contraction, graph transversal, parameterized complexity, vertex cover, feedback vertex set, odd cycle transversal} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

In this paper, we consider the following problem: given a connected graph G, can we reduce the domination number of G by one by using only one edge contraction? We show that the problem is NP-hard when restricted to {P_6,P_4+P_2}-free graphs and that it is coNP-hard when restricted to subcubic claw-free graphs and 2P_3-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on H-free graphs when H is connected.

Esther Galby, Paloma T. Lima, and Bernard Ries. Blocking Dominating Sets for H-Free Graphs via Edge Contractions. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galby_et_al:LIPIcs.ISAAC.2019.21, author = {Galby, Esther and Lima, Paloma T. and Ries, Bernard}, title = {{Blocking Dominating Sets for H-Free Graphs via Edge Contractions}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.21}, URN = {urn:nbn:de:0030-drops-115171}, doi = {10.4230/LIPIcs.ISAAC.2019.21}, annote = {Keywords: domination number, blocker problem, H-free graphs} }

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

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by chi_b(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on chi_b(G): The maximum degree Delta(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i-1. We obtain a dichotomy result for all fixed k in N when k is close to one of the two above mentioned upper bounds. Concretely, we show that if k in {Delta(G) + 1 - p, m(G) - p}, the problem is polynomial-time solvable whenever p in {0, 1} and, even when k = 3, it is NP-complete whenever p >= 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Delta(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Delta(G). Second, we show that b-Coloring{} is FPT parameterized by Delta(G) + l_k(G), where l_k(G) denotes the number of vertices of degree at least k.

Lars Jaffke and Paloma T. Lima. A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2019.34, author = {Jaffke, Lars and Lima, Paloma T.}, title = {{A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {34:1--34:13}, 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.2019.34}, URN = {urn:nbn:de:0030-drops-109784}, doi = {10.4230/LIPIcs.MFCS.2019.34}, annote = {Keywords: b-Coloring, b-Chromatic Number} }

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

In this paper, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k >= 0? We show that for k <= 2, the problem is coNP-hard. We further prove that for k=1, the problem is W[1]-hard parameterized by the size of a minimum dominating set plus the mim-width of the input graph, and that it remains NP-hard when restricted to P_9-free graphs, bipartite graphs and {C_3,...,C_{l}}-free graphs for any l >= 3. Finally, we show that for any k >= 1, the problem is polynomial-time solvable for P_5-free graphs and that it can be solved in FPT-time and XP-time when parameterized by tree-width and mim-width, respectively.

Esther Galby, Paloma T. Lima, and Bernard Ries. Reducing the Domination Number of Graphs via Edge Contractions. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galby_et_al:LIPIcs.MFCS.2019.41, author = {Galby, Esther and Lima, Paloma T. and Ries, Bernard}, title = {{Reducing the Domination Number of Graphs via Edge Contractions}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {41:1--41:13}, 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.2019.41}, URN = {urn:nbn:de:0030-drops-109856}, doi = {10.4230/LIPIcs.MFCS.2019.41}, annote = {Keywords: domination number, blocker problem, graph classes} }

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

Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors. We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. We study the problem of deciding whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. Although edge-colorings have been studied extensively under similar constraints, there are significantly fewer results on the vertex variant that we consider. In particular, its complexity on structured graph classes was explicitly posed as an open question.
We show that the problem remains NP-complete even on bipartite apex graphs and on split graphs. The former can be seen as a first step in the direction of studying the complexity of rainbow coloring on sparse graphs, an open problem which has attracted attention but limited progress. We also give hardness of approximation results for both bipartite and split graphs. To complement the negative results, we show that bipartite permutation graphs, interval graphs, and block graphs can be rainbow vertex-connected optimally in polynomial time.

Pinar Heggernes, Davis Issac, Juho Lauri, Paloma T. Lima, and Erik Jan van Leeuwen. Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 83:1-83:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{heggernes_et_al:LIPIcs.MFCS.2018.83, author = {Heggernes, Pinar and Issac, Davis and Lauri, Juho and Lima, Paloma T. and van Leeuwen, Erik Jan}, title = {{Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {83:1--83:13}, 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.2018.83}, URN = {urn:nbn:de:0030-drops-96657}, doi = {10.4230/LIPIcs.MFCS.2018.83}, annote = {Keywords: Rainbow coloring, graph classes, polynomial-time algorithms, approximation algorithms} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the Strong F-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In Strong F-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to destroy that strong copy of F in G.
We study Strong F-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when F =P_3. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization k - mu(G), where mu(G) is the maximum matching size of G. We conclude with some results on the parameterization of Strong F-closure by the number of edges of G that are not selected as strong.

Petr A. Golovach, Pinar Heggernes, Athanasios L. Konstantinidis, Paloma T. Lima, and Charis Papadopoulos. Parameterized Aspects of Strong Subgraph Closure. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{golovach_et_al:LIPIcs.SWAT.2018.23, author = {Golovach, Petr A. and Heggernes, Pinar and Konstantinidis, Athanasios L. and Lima, Paloma T. and Papadopoulos, Charis}, title = {{Parameterized Aspects of Strong Subgraph Closure}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {23:1--23:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.23}, URN = {urn:nbn:de:0030-drops-88490}, doi = {10.4230/LIPIcs.SWAT.2018.23}, annote = {Keywords: Strong triadic closure, Parameterized complexity, Forbidden subgraphs} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The Pi-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property Pi. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties Pi, even when parameterized by k+t. We show that the situation changes when we are looking for a connected induced subgraph satisfying Pi. In particular, we show that the Connected Secluded Pi-Subgraph problem is FPT when parameterized by just t for many important graph properties Pi.

Petr A. Golovach, Pinar Heggernes, Paloma T. Lima, and Pedro Montealegre. Finding Connected Secluded Subgraphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{golovach_et_al:LIPIcs.IPEC.2017.18, author = {Golovach, Petr A. and Heggernes, Pinar and Lima, Paloma T. and Montealegre, Pedro}, title = {{Finding Connected Secluded Subgraphs}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {18:1--18:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.18}, URN = {urn:nbn:de:0030-drops-85623}, doi = {10.4230/LIPIcs.IPEC.2017.18}, annote = {Keywords: Secluded subgraph, forbidden subgraphs, parameterized complexity} }

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