Curious crossing-critical edges -- variations on an example of \v{S}ir\'a\v{n}

TL;DR

This paper explores complex relationships between crossing-critical edges and Kuratowski subgraphs, providing examples that challenge intuitive assumptions about their intersections in graph drawings.

Contribution

It presents novel examples of graphs illustrating surprising behaviors of crossing-critical edges relative to Kuratowski subgraphs, expanding understanding of crossing number properties.

Findings

An edge crossed in every optimal drawing but not in any Kuratowski subgraph.

An edge in every Kuratowski subgraph but not crossed in any optimal drawing.

A crossing-critical edge not in any Kuratowski subgraph and not crossed in optimal drawings.

Abstract

Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we present the following examples: a graph with an edge that is crossed in every optimal drawing of the graph, but the edge is not in any Kuratowski subgraph of the graph; a graph with an edge that is in every Kuratowski subgraph but is not crossed in any optimal drawing of the graph; and a graph with a crossing-critical edge that is not present in any Kuratowski subgraph and is not crossed in any optimal drawing of the graph. F\'ary's theorem implies that the Kuratowski subgraphs are the only obstructions to a graph having a crossing-free drawing with all edges drawn as straight lines. The three example graphs given also hold if we restrict drawings to only have straight line edges, and thus also apply to the rectilinear crossing number.