Making a graph crossing-critical by multiplying its edges

TL;DR

This paper investigates how multiplying edges in certain graphs can induce crossing-criticality, revealing structural conditions under which noncritical graphs can become crossing-critical through edge multiplication.

Contribution

It demonstrates that nonplanar, sufficiently connected graphs derived from cubic polyhedral graphs can be made crossing-critical by multiplying edges, highlighting a method to induce crossing-criticality.

Findings

Edge multiplication can induce crossing-criticality in certain noncritical graphs.

Sufficient connectivity and specific construction from cubic polyhedral graphs are key.

Crossing-criticality can be structurally induced, not just inherent.

Abstract

A graph is crossing-critical if the removal of any of its edges decreases its crossing number. This work is motivated by the following question: to what extent is crossing- criticality a property that is inherent to the structure of a graph, and to what extent can it be induced on a noncritical graph by multiplying (all or some of) its edges? It is shown that if a nonplanar graph G is obtained by adding an edge to a cubic polyhedral graph, and G is sufficiently connected, then G can be made crossing-critical by a suitable multiplication of edges.