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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

A map is a 2-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. When the underlying surface is orientable, every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map, parametrized by the genus of the underlying surface. The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover.

Ken-ichi Kawarabayashi, Bojan Mohar, Roman Nedela, and Peter Zeman. Automorphisms and Isomorphisms of Maps in Linear Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 86:1-86:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kawarabayashi_et_al:LIPIcs.ICALP.2021.86, author = {Kawarabayashi, Ken-ichi and Mohar, Bojan and Nedela, Roman and Zeman, Peter}, title = {{Automorphisms and Isomorphisms of Maps in Linear Time}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {86:1--86:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.86}, URN = {urn:nbn:de:0030-drops-141558}, doi = {10.4230/LIPIcs.ICALP.2021.86}, annote = {Keywords: maps on surfaces, automorphisms, isomorphisms, algorithm} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c >= 13 and d >= 1, we give first explicit constructions of c-crossing-critical graphs containing a vertex of degree greater than d. We also show that such unbounded degree constructions do not exist for c <=12, precisely, that there exists a constant D such that every c-crossing-critical graph with c <=12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of c-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c <=12.

Drago Bokal, Zdeněk Dvořák, Petr Hliněný, Jesús Leaños, Bojan Mohar, and Tilo Wiedera. Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bokal_et_al:LIPIcs.SoCG.2019.14, author = {Bokal, Drago and Dvo\v{r}\'{a}k, Zden\v{e}k and Hlin\v{e}n\'{y}, Petr and Lea\~{n}os, Jes\'{u}s and Mohar, Bojan and Wiedera, Tilo}, title = {{Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c \langle= 12}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.14}, URN = {urn:nbn:de:0030-drops-104183}, doi = {10.4230/LIPIcs.SoCG.2019.14}, annote = {Keywords: graph drawing, crossing number, crossing-critical, zip product} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the topological crossing number of a graph, but is more general.
The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed.
Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H' of a graph G', and an embedding of H' on a surface S, can that embedding be extended to an embedding of G' on S? Such problems can be solved, in turn, using a key component in Mohar's algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).

Éric Colin de Verdière, Thomas Magnard, and Bojan Mohar. Embedding Graphs into Two-Dimensional Simplicial Complexes. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{verdiere_et_al:LIPIcs.SoCG.2018.27, author = {Verdi\`{e}re, \'{E}ric Colin de and Magnard, Thomas and Mohar, Bojan}, title = {{Embedding Graphs into Two-Dimensional Simplicial Complexes}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.27}, URN = {urn:nbn:de:0030-drops-87401}, doi = {10.4230/LIPIcs.SoCG.2018.27}, annote = {Keywords: computational topology, embedding, simplicial complex, graph, surface} }

Document

**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_{3,3}, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the c-crossing-critical graphs of at most given order n in polynomial time per each generated graph.

Zdenek Dvorák, Petr Hlinený, and Bojan Mohar. Structure and Generation of Crossing-Critical Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.SoCG.2018.33, author = {Dvor\'{a}k, Zdenek and Hlinen\'{y}, Petr and Mohar, Bojan}, title = {{Structure and Generation of Crossing-Critical Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.33}, URN = {urn:nbn:de:0030-drops-87460}, doi = {10.4230/LIPIcs.SoCG.2018.33}, annote = {Keywords: crossing number, crossing-critical, path-width, exhaustive generation} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We present a polynomial-time 9/7-approximation algorithm for the graphic TSP for cubic graphs, which improves the previously best approximation factor of 1.3 for 2-connected cubic graphs and drops the requirement of 2-connectivity at the same time. To design our algorithm, we prove that every simple 2-connected cubic n-vertex graph contains a spanning closed walk of length at most 9n/7-1, and that such a walk can be found in polynomial time.

Zdenek Dvorák, Daniel Král, and Bojan Mohar. Graphic TSP in Cubic Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2017.27, author = {Dvor\'{a}k, Zdenek and Kr\'{a}l, Daniel and Mohar, Bojan}, title = {{Graphic TSP in Cubic Graphs}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {27:1--27:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.27}, URN = {urn:nbn:de:0030-drops-70068}, doi = {10.4230/LIPIcs.STACS.2017.27}, annote = {Keywords: Graphic TSP, approximation algorithms, cubic graphs} }

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