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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In this paper, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths 𝒫 and 𝒬 connecting the same set of terminal pairs, we aim to determine whether 𝒫 can be transformed to 𝒬 by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when k = 2. On the other hand, we prove that, when the graph is embedded on a plane and all paths in 𝒫 and 𝒬 connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s-t paths as a variant. We show that the disjoint s-t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.

Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Rerouting Planar Curves and Disjoint Paths. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 81:1-81:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ito_et_al:LIPIcs.ICALP.2023.81, author = {Ito, Takehiro and Iwamasa, Yuni and Kakimura, Naonori and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio and Ozeki, Kenta}, title = {{Rerouting Planar Curves and Disjoint Paths}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {81:1--81:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.81}, URN = {urn:nbn:de:0030-drops-181339}, doi = {10.4230/LIPIcs.ICALP.2023.81}, annote = {Keywords: Disjoint paths, combinatorial reconfiguration, planar graphs} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In 1973, Fisk proved that any 4-coloring of a 3-colorable triangulation of the 2-sphere can be obtained from any 3-coloring by a sequence of Kempe-changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe-change, there exists a 4-coloring that cannot be obtained from any 3-coloring.
In this paper, we present a linear-time checkable characterization of a 4-coloring of a 3-colorable triangulation of the 2-sphere that can be obtained from a 3-coloring by a sequence of recoloring operations at single vertices. In addition, we develop a quadratic-time algorithm to find such a recoloring sequence if it exists; our proof implies that we can always obtain a quadratic length recoloring sequence. We also present a linear-time checkable criterion for a 3-colorable triangulation of the 2-sphere that all 4-colorings can be obtained from a 3-coloring by such a sequence. Moreover, we consider a high-dimensional setting. As a natural generalization of our first result, we obtain a polynomial-time checkable characterization of a k-coloring of a (k-1)-colorable triangulation of the (k-2)-sphere that can be obtained from a (k-1)-coloring by a sequence of recoloring operations at single vertices and the corresponding algorithmic result. Furthermore, we show that the problem of deciding whether, for given two (k+1)-colorings of a (k-1)-colorable triangulation of the (k-2)-sphere, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed k ≥ 4. Our results above can be rephrased as new results on the computational problems named k-Recoloring and Connectedness of k-Coloring Reconfiguration Graph, which are fundamental problems in the field of combinatorial reconfiguration.

Takehiro Ito, Yuni Iwamasa, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Reconfiguration of Colorings in Triangulations of the Sphere. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ito_et_al:LIPIcs.SoCG.2023.43, author = {Ito, Takehiro and Iwamasa, Yuni and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio and Ozeki, Kenta}, title = {{Reconfiguration of Colorings in Triangulations of the Sphere}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {43:1--43:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.43}, URN = {urn:nbn:de:0030-drops-178936}, doi = {10.4230/LIPIcs.SoCG.2023.43}, annote = {Keywords: Graph coloring, Triangulation of the sphere, Combinatorial reconfiguration} }

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