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On the Independence Number of 1-Planar Graphs

Authors Therese Biedl, Prosenjit Bose, Babak Miraftab

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • 1-planar graph
  • independent set
  • minimum degree

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Abstract

An independent set in a graph is a set of vertices where no two vertices are adjacent to each other. A maximum independent set is the largest possible independent set that can be formed within a given graph G. The cardinality of this set is referred to as the independence number of G. This paper investigates the independence number of 1-planar graphs, a subclass of graphs defined by drawings in the Euclidean plane where each edge can have at most one crossing point. Borodin establishes a tight upper bound of six for the chromatic number of every 1-planar graph G, leading to a corresponding lower bound of n/6 for the independence number, where n is the number of vertices of G. In contrast, the upper bound for the independence number in 1-planar graphs is less studied. This paper addresses this gap by presenting upper bounds based on the minimum degree δ. A comprehensive table summarizes these upper bounds for various δ values, providing insights into achievable independence numbers under different conditions.

Cite As Get BibTex

Therese Biedl, Prosenjit Bose, and Babak Miraftab. On the Independence Number of 1-Planar Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.13

Author Details

Therese Biedl
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Babak Miraftab
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

  • Biedl, Therese: Supported by NSERC; FRN RGPIN-2020-03958.
  • Bose, Prosenjit: Supported by NSERC.
  • Miraftab, Babak: Supported by NSERC.

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