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Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor

Authors Vida Dujmović , Camille La Rose

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graphs and surfaces
  • Theory of computation → Computational geometry
Keywords
  • (rectilinear) crossing number
  • graph minors
  • maximum degree
  • clique-sums

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Abstract

The rectilinear crossing number of G is the minimum number of crossings in a straight-line drawing of G. A single-crossing graph is a graph whose crossing number is at most one. We prove that every n-vertex graph G that excludes a single-crossing graph as a minor has rectilinear crossing number O(Δ n), where Δ is the maximum degree of G. This dependence on n and Δ is best possible. The result applies, for example, to K₅-minor-free graphs, and bounded treewidth graphs. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth as well as bounded degree K_{3,3}-minor-free graphs. In the case of bounded treewidth graphs, our O(Δ n) result is again tight and it improves on the previous best known bound of O(Δ² n) by Wood and Telle, 2007.

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Vida Dujmović and Camille La Rose. Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.37

Author Details

Vida Dujmović
  • University of Ottawa, Canada
Camille La Rose
  • University of Ottawa, Canada

Funding

  • Dujmović, Vida: Funded by NSERC and a University of Ottawa Research Chair.
  • La Rose, Camille: Funded by NSERC.

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