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

Let Π₁, Π₂, …, Π_c be graph properties for a fixed integer c. Then, (Π₁, Π₂, …, Π_c)-Partition is the problem of asking whether the vertex set of a given graph can be partitioned into c subsets V₁, V₂, …, V_c such that the subgraph induced by V_i satisfies the graph property Π_i for every i ∈ {1,2, …, c}. Minimization and parameterized variants of (Π₁, Π₂, …, Π_c)-Partition have been studied for several specific graph properties, where the size of the vertex subset V₁ satisfying Π₁ is minimized or taken as a parameter. In this paper, we first show that the minimization variant is hard to approximate for any nontrivial additive hereditary graph properties, unless c = 2 and both Π₁ and Π₂ are classes of edgeless graphs. We then give FPT algorithms for the parameterized variant when restricted to the case where c = 2, Π₁ is a hereditary graph property, and Π₂ is the class of acyclic graphs.

Yuma Tamura, Takehiro Ito, and Xiao Zhou. Minimization and Parameterized Variants of Vertex Partition Problems on Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{tamura_et_al:LIPIcs.ISAAC.2020.40, author = {Tamura, Yuma and Ito, Takehiro and Zhou, Xiao}, title = {{Minimization and Parameterized Variants of Vertex Partition Problems on Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {40:1--40:13}, 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.2020.40}, URN = {urn:nbn:de:0030-drops-133844}, doi = {10.4230/LIPIcs.ISAAC.2020.40}, annote = {Keywords: Graph Algorithms, Approximability, Fixed-Parameter Tractability, Vertex Partition Problem, Feedback Vertex Set Problem} }

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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 problem of deciding if there is a transformation between two given minimum Steiner trees of an unweighted graph such that each transformation step respects a prescribed reconfiguration rule and results in another minimum Steiner tree of the graph. We consider two reconfiguration rules, both of which exchange a single vertex at a time, and generalize the known reconfiguration problem for shortest paths in an unweighted graph. This generalization implies that our problems under both reconfiguration rules are PSPACE-complete for bipartite graphs. We thus study the problems with respect to graph classes, and give some boundaries between the polynomial-time solvable and PSPACE-complete cases.

Haruka Mizuta, Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Reconfiguration of Minimum Steiner Trees via Vertex Exchanges. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 79:1-79:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mizuta_et_al:LIPIcs.MFCS.2019.79, author = {Mizuta, Haruka and Hatanaka, Tatsuhiko and Ito, Takehiro and Zhou, Xiao}, title = {{Reconfiguration of Minimum Steiner Trees via Vertex Exchanges}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {79:1--79:11}, 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.2019.79}, URN = {urn:nbn:de:0030-drops-110234}, doi = {10.4230/LIPIcs.MFCS.2019.79}, annote = {Keywords: Combinatorial reconfiguration, Graph algorithms, Steiner tree} }

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

Coloring reconfiguration is one of the most well-studied reconfiguration problems. In the problem, we are given two (vertex-)colorings of a graph using at most k colors, and asked to determine whether there exists a transformation between them by recoloring only a single vertex at a time, while maintaining a k-coloring throughout. It is known that this problem is solvable in linear time for any graph if k <=3, while is PSPACE-complete for a fixed k >= 4. In this paper, we further investigate the problem from the viewpoint of recolorability constraints, which forbid some pairs of colors to be recolored directly. More specifically, the recolorability constraint is given in terms of an undirected graph R such that each node in R corresponds to a color, and each edge in R represents a pair of colors that can be recolored directly. In this paper, we give a linear-time algorithm to solve the problem under such a recolorability constraint if R is of maximum degree at most two. In addition, we show that the minimum number of recoloring steps required for a desired transformation can be computed in linear time for a yes-instance. We note that our results generalize the known positive ones for coloring reconfiguration.

Hiroki Osawa, Akira Suzuki, Takehiro Ito, and Xiao Zhou. Algorithms for Coloring Reconfiguration Under Recolorability Constraints. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{osawa_et_al:LIPIcs.ISAAC.2018.37, author = {Osawa, Hiroki and Suzuki, Akira and Ito, Takehiro and Zhou, Xiao}, title = {{Algorithms for Coloring Reconfiguration Under Recolorability Constraints}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {37:1--37:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.37}, URN = {urn:nbn:de:0030-drops-99850}, doi = {10.4230/LIPIcs.ISAAC.2018.37}, annote = {Keywords: combinatorial reconfiguration, graph algorithm, graph coloring} }

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

For an integer k \ge 1, k-coloring reconfiguration is one of the most well-studied reconfiguration problems, defined as follows: In the problem, we are given two (vertex-)colorings of a graph using k colors, and asked to transform one into the other by recoloring only one vertex at a time, while at all times maintaining a proper coloring. The problem is known to be PSPACE-complete if k \ge 4, and solvable for any graph in polynomial time if k \le 3. In this paper, we introduce a recolorability constraint on the k colors, which forbids some pairs of colors to be recolored directly. The recolorability constraint is given in terms of an undirected graph R such that each node in R corresponds to a color and each edge in R represents a pair of colors that can be recolored directly. We study the hardness of the problem based on the structure of recolorability constraints R. More specifically, we prove that the problem is PSPACE-complete if R is of maximum degree at least four, or has a connected component containing more than one cycle.

Hiroki Osawa, Akira Suzuki, Takehiro Ito, and Xiao Zhou. Complexity of Coloring Reconfiguration under Recolorability Constraints. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{osawa_et_al:LIPIcs.ISAAC.2017.62, author = {Osawa, Hiroki and Suzuki, Akira and Ito, Takehiro and Zhou, Xiao}, title = {{Complexity of Coloring Reconfiguration under Recolorability Constraints}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {62:1--62:13}, 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.2017.62}, URN = {urn:nbn:de:0030-drops-82588}, doi = {10.4230/LIPIcs.ISAAC.2017.62}, annote = {Keywords: combinatorial reconfiguration, graph coloring, PSPACE-complete} }

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

Let G be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. For two given list colorings of G, we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant k. In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by k and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant which computes the length of a shortest transformation when parameterized by k and the size of a minimum vertex cover of an input graph. As corollaries, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only k is taken as a parameter. On the other hand, we prove that the problem is W[1]-hard when parameterized only by the size of a minimum vertex cover of an input graph.

Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Parameterized Complexity of the List Coloring Reconfiguration Problem with Graph Parameters. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hatanaka_et_al:LIPIcs.MFCS.2017.51, author = {Hatanaka, Tatsuhiko and Ito, Takehiro and Zhou, Xiao}, title = {{Parameterized Complexity of the List Coloring Reconfiguration Problem with Graph Parameters}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {51:1--51:13}, 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.2017.51}, URN = {urn:nbn:de:0030-drops-81060}, doi = {10.4230/LIPIcs.MFCS.2017.51}, annote = {Keywords: combinatorial reconfiguration, fixed-parameter tractability, graph algorithm, list coloring, W\lbrack1\rbrack-hardness} }

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