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Invited Talk

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

Kempe changes were introduced in 1879 in an attempt to prove the 4-colour theorem. They are a convenient if not crucial tool to prove various colouring theorems. Here, we consider how to navigate from a colouring to another through Kempe changes. When is it possible? How fast?

Marthe Bonamy. Exploring the Space of Colourings with Kempe Changes (Invited Talk). In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bonamy:LIPIcs.MFCS.2023.1, author = {Bonamy, Marthe}, title = {{Exploring the Space of Colourings with Kempe Changes}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {1:1--1:2}, 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.2023.1}, URN = {urn:nbn:de:0030-drops-185350}, doi = {10.4230/LIPIcs.MFCS.2023.1}, annote = {Keywords: Graph theory, graph coloring, reconfiguration} }

Document

**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a (5-ε)-approximation, for any ε > 0. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.

Marthe Bonamy, Linda Cook, Carla Groenland, and Alexandra Wesolek. A Tight Local Algorithm for the Minimum Dominating Set Problem in Outerplanar Graphs. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bonamy_et_al:LIPIcs.DISC.2021.13, author = {Bonamy, Marthe and Cook, Linda and Groenland, Carla and Wesolek, Alexandra}, title = {{A Tight Local Algorithm for the Minimum Dominating Set Problem in Outerplanar Graphs}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.13}, URN = {urn:nbn:de:0030-drops-148159}, doi = {10.4230/LIPIcs.DISC.2021.13}, annote = {Keywords: Outerplanar graphs, dominating set, LOCAL model, constant-factor approximation algorithm} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

A k-coloring of a graph maps each vertex of the graph to a color in {1, 2, …, k}, such that no two adjacent vertices receive the same color. Given a k-coloring of a graph, a Kempe change produces a new k-coloring by swapping the colors in a bicolored connected component. We investigate the complexity of finding the smallest number of Kempe changes needed to transform a given k-coloring into another given k-coloring. We show that this problem admits a polynomial-time dynamic programming algorithm on path graphs, which turns out to be highly non-trivial. Furthermore, the problem is NP-hard even on star graphs and we show that on such graphs it admits a constant-factor approximation algorithm and is fixed-parameter tractable when parameterized by the number k of colors. The hardness result as well as the algorithmic results are based on the notion of a canonical transformation.

Marthe Bonamy, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Moritz Mühlenthaler, Akira Suzuki, and Kunihiro Wasa. Shortest Reconfiguration of Colorings Under Kempe Changes. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonamy_et_al:LIPIcs.STACS.2020.35, author = {Bonamy, Marthe and Heinrich, Marc and Ito, Takehiro and Kobayashi, Yusuke and Mizuta, Haruka and M\"{u}hlenthaler, Moritz and Suzuki, Akira and Wasa, Kunihiro}, title = {{Shortest Reconfiguration of Colorings Under Kempe Changes}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {35:1--35:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.35}, URN = {urn:nbn:de:0030-drops-118961}, doi = {10.4230/LIPIcs.STACS.2020.35}, annote = {Keywords: Combinatorial Reconfiguration, Graph Algorithms, Graph Coloring, Kempe Equivalence} }

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

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.

Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz Mühlenthaler, and Kunihiro Wasa. The Perfect Matching Reconfiguration Problem. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bonamy_et_al:LIPIcs.MFCS.2019.80, author = {Bonamy, Marthe and Bousquet, Nicolas and Heinrich, Marc and Ito, Takehiro and Kobayashi, Yusuke and Mary, Arnaud and M\"{u}hlenthaler, Moritz and Wasa, Kunihiro}, title = {{The Perfect Matching Reconfiguration Problem}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {80:1--80:14}, 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.2019.80}, URN = {urn:nbn:de:0030-drops-110248}, doi = {10.4230/LIPIcs.MFCS.2019.80}, annote = {Keywords: Combinatorial Reconfiguration, Graph Algorithms, Perfect Matching} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem.

Marthe Bonamy, Oscar Defrain, Marc Heinrich, and Jean-Florent Raymond. Enumerating Minimal Dominating Sets in Triangle-Free Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bonamy_et_al:LIPIcs.STACS.2019.16, author = {Bonamy, Marthe and Defrain, Oscar and Heinrich, Marc and Raymond, Jean-Florent}, title = {{Enumerating Minimal Dominating Sets in Triangle-Free Graphs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {16:1--16:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.16}, URN = {urn:nbn:de:0030-drops-102557}, doi = {10.4230/LIPIcs.STACS.2019.16}, annote = {Keywords: Enumeration algorithms, output-polynomial algorithms, minimal dominating set, triangle-free graphs, split graphs} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step?
We introduce the notion of distributed recoloring: The input graph represents a network of computers that needs to be recolored. Initially, each node is aware of its own input color and target color. The nodes can exchange messages with each other, and eventually each node has to stop and output its own recoloring schedule, indicating when and how the node changes its color. The recoloring schedules have to be globally consistent so that the graph remains properly colored at each point, and we require that adjacent nodes do not change their colors simultaneously.
We are interested in the following questions: How many communication rounds are needed (in the deterministic LOCAL model of distributed computing) to find a recoloring schedule? What is the length of the recoloring schedule? And how does the picture change if we can use extra colors to make recoloring easier?
The main contributions of this work are related to distributed recoloring with one extra color in the following graph classes: trees, 3-regular graphs, and toroidal grids.

Marthe Bonamy, Paul Ouvrard, Mikaël Rabie, Jukka Suomela, and Jara Uitto. Distributed Recoloring. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bonamy_et_al:LIPIcs.DISC.2018.12, author = {Bonamy, Marthe and Ouvrard, Paul and Rabie, Mika\"{e}l and Suomela, Jukka and Uitto, Jara}, title = {{Distributed Recoloring}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.12}, URN = {urn:nbn:de:0030-drops-98012}, doi = {10.4230/LIPIcs.DISC.2018.12}, annote = {Keywords: Distributed Systems, Graph Algorithms, Local Computations} }

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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 87, 25th Annual European Symposium on Algorithms (ESA 2017)

In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b=1 case) is equivalent to finding a homomorphism to the Kneser graph KG_{a,b}, and gives relaxations approaching the fractional chromatic number.
We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with running time f(b) * 2^{o(log b) n}, for any computable f(b), unless the Exponential Time Hypothesis (ETH) fails. A (b+1)^n * poly(n)-time algorithm due to Nederlof [2008] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2^O(n+h) algorithm unless ETH fails, even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [SODA 2016].
The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [Canad. Math. Bull., 1965], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the running time of the algorithms of Abasi et al. [MFCS 2014] and of Gabizon et al. [ESA 2015] for the r-monomial detection problem are optimal under ETH.

Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. Tight Lower Bounds for the Complexity of Multicoloring. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.ESA.2017.18, author = {Bonamy, Marthe and Kowalik, Lukasz and Pilipczuk, Michal and Socala, Arkadiusz and Wrochna, Marcin}, title = {{Tight Lower Bounds for the Complexity of Multicoloring}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.18}, URN = {urn:nbn:de:0030-drops-78527}, doi = {10.4230/LIPIcs.ESA.2017.18}, annote = {Keywords: multicoloring, Kneser graph homomorphism, ETH lower bound} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

In the k-Leaf Out-Branching and k-Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is to determine the existence of an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k^2) vertices are known on general graphs. In this work we show that k-Leaf Out-Branching admits a kernel with O(k) vertices on H-minor-free graphs, for any fixed H, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.

Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, and Arkadiusz Socala. Linear Kernels for Outbranching Problems in Sparse Digraphs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 199-211, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bonamy_et_al:LIPIcs.IPEC.2015.199, author = {Bonamy, Marthe and Kowalik, Lukasz and Pilipczuk, Michal and Socala, Arkadiusz}, title = {{Linear Kernels for Outbranching Problems in Sparse Digraphs}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {199--211}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.199}, URN = {urn:nbn:de:0030-drops-55839}, doi = {10.4230/LIPIcs.IPEC.2015.199}, annote = {Keywords: FPT algorithm, kernelization, outbranching, sparse graphs} }