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DOI: 10.4230/LIPIcs.MFCS.2024.66

Twin-Width of Graphs on Surfaces

Authors: Daniel Kráľ, Kristýna Pekárková, and Kenny Štorgel

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract

Twin-width is a width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. Hliněný and Jedelský [ICALP'23] showed that every planar graph has twin-width at most 8. We prove that the twin-width of every graph embeddable in a surface of Euler genus g is at most 18√{47g} + O(1), which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus g that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size max{6, 32g-37}.

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Daniel Kráľ, Kristýna Pekárková, and Kenny Štorgel. Twin-Width of Graphs on Surfaces. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kral_et_al:LIPIcs.MFCS.2024.66,
  author =	{Kr\'{a}\v{l}, Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na and \v{S}torgel, Kenny},
  title =	{{Twin-Width of Graphs on Surfaces}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{66:1--66:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.66},
  URN =		{urn:nbn:de:0030-drops-206226},
  doi =		{10.4230/LIPIcs.MFCS.2024.66},
  annote =	{Keywords: twin-width, graphs on surfaces, fixed parameter tractability}
}

Document

DOI: 10.4230/LIPIcs.STACS.2017.27

Graphic TSP in Cubic Graphs

Authors: Zdenek Dvorák, Daniel Král, and Bojan Mohar

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

Abstract

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.

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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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Twin-Width of Graphs on Surfaces

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Graphic TSP in Cubic Graphs

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