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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.23
On Computing the Vertex Connectivity of 1-Plane Graphs

Authors: Therese Biedl and Karthik Murali

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

Abstract
A graph is called 1-plane if it has an embedding in the plane where each edge is crossed at most once by another edge. A crossing of a 1-plane graph is called an ×-crossing if there are no other edges connecting the endpoints of the crossing (apart from the crossing pair of edges). In this paper, we show how to compute the vertex connectivity of a 1-plane graph G without ×-crossings in linear time. To do so, we show that for any two vertices u,v in a minimum separating set S, the distance between u and v in an auxiliary graph Λ(G) (obtained by planarizing G and then inserting into each face a new vertex adjacent to all vertices of the face) is small. It hence suffices to search for a minimum separating set in various subgraphs Λ_i of Λ(G) with small diameter. Since Λ(G) is planar, the subgraphs Λ_i have small treewidth. Each minimum separating set S then gives rise to a partition of Λ_i into three vertex sets with special properties; such a partition can be found via Courcelle’s theorem in linear time.
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Therese Biedl and Karthik Murali. On Computing the Vertex Connectivity of 1-Plane Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{biedl_et_al:LIPIcs.ICALP.2023.23,
  author =	{Biedl, Therese and Murali, Karthik},
  title =	{{On Computing the Vertex Connectivity of 1-Plane Graphs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{23:1--23:16},
  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.2023.23},
  URN =		{urn:nbn:de:0030-drops-180753},
  doi =		{10.4230/LIPIcs.ICALP.2023.23},
  annote =	{Keywords: 1-Planar Graph, Connectivity, Linear Time, Treewidth}
}
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