On the decay of crossing numbers of sparse graphs

TL;DR

This paper investigates how the crossing number of sparse graphs decreases when edges are removed, extending previous dense graph results and exploring the relationship with expected crossing numbers.

Contribution

It extends the understanding of crossing number decay to large sparse graphs without edge inflation, complementing known results for dense graphs.

Findings

Established decay bounds for crossing numbers in sparse graphs.

Connected decay behavior to the concept of expected crossing numbers.

Provided insights into the structural properties affecting crossing number reduction.

Abstract

Richter and Thomassen proved that every graph has an edge $e$ such that the crossing number $\ucr(G-e)$ of $G-e$ is at least $(2/5)\ucr(G) - O(1)$. Fox and Cs. T\'oth proved that dense graphs have large sets of edges (proportional in the total number of edges) whose removal leaves a graph with crossing number proportional to the crossing number of the original graph; this result was later strenghtened by \v{C}ern\'{y}, Kyn\v{c}l and G. T\'oth. These results make our understanding of the {decay} of crossing numbers in dense graphs essentially complete. In this paper we prove a similar result for large sparse graphs in which the number of edges is not artificially inflated by operations such as edge subdivisions. We also discuss the connection between the decay of crossing numbers and expected crossing numbers, a concept recently introduced by Mohar and Tamon.