A Note on the Maximum Rectilinear Crossing Number of Spiders

TL;DR

This paper investigates the maximum rectilinear crossing number of spider graphs, providing bounds and exact values for subdivided stars, advancing understanding of crossings in tree-like graphs.

Contribution

It offers new bounds and exact calculations for the maximum rectilinear crossing number of subdivided star graphs, a step towards general trees.

Findings

Established bounds for spiders' crossing numbers

Calculated exact crossing numbers for subdivided stars

Method applicable to broader classes of trees

Abstract

The maximum rectilinear crossing number of a graph $G$ is the maximum number of crossings in a good straight-line drawing of $G$ in the plane. In a good drawing any two edges intersect in at most one point (counting endpoints), no three edges have an interior point in common, and edges do not contain vertices in their interior. A spider is a subdivision of $K_{1,k}$. We provide both upper and lower bounds for the maximum rectilinear crossing number of spiders. While there are not many results on the maximum rectilinear crossing numbers of infinite families of graphs, our methods can be used to find the exact maximum rectilinear crossing number of $K_{1,k}$ where each edge is subdivided exactly once. This is a first step towards calculating the maximum rectilinear crossing number of arbitrary trees.