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A Parameterized Algorithm for Vertex and Edge Connectivity of Embedded Graphs

Authors Therese Biedl , Prosenjit Bose , Karthik Murali

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Author Details

Therese Biedl
  • University of Waterloo, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Karthik Murali
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

Supported by NSERC.

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Therese Biedl, Prosenjit Bose, and Karthik Murali. A Parameterized Algorithm for Vertex and Edge Connectivity of Embedded Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.24

Abstract

The problem of computing vertex and edge connectivity of a graph are classical problems in algorithmic graph theory. The focus of this paper is on computing these parameters for graphs drawn on the plane. A typical example of such graphs are planar graphs which can be embedded without any crossings. It has long been known that vertex and edge connectivity of planar embedded graphs can be computed in linear time. Very recently, Biedl and Murali extended the techniques from planar graphs to 1-plane graphs without ×-crossings, i.e., crossings whose endpoints induce a matching. While the tools used were novel, they were highly tailored to 1-plane graphs, and do not provide much leeway for further extension. In this paper, we develop alternate techniques that are simpler, have wider applications to near-planar graphs, and can be used to test both vertex and edge connectivity. Our technique works for all those embedded graphs where any pair of crossing edges are connected by a path that, roughly speaking, can be covered with few cells of the drawing. Important examples of such graphs include optimal 2-planar and optimal 3-planar graphs, d-map graphs, d-framed graphs, graphs with bounded crossing number, and k-plane graphs with bounded number of ×-crossings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Vertex Connectivity
  • Edge Connectivity
  • 1-planar
  • k-planar
  • Linear-time

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References

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