Minor crossing number is additive over arbitrary cuts

TL;DR

This paper proves that the minor crossing number is additive over arbitrary cuts and extends the additivity of crossing number for small edge cuts, with implications for graph crossing number computation and properties.

Contribution

It establishes the additivity of minor crossing number over arbitrary edge cuts and extends crossing number additivity results to small cuts and various surfaces.

Findings

Crossing number is additive over edge cuts of size 0 to 3.

Minor crossing number is additive over edge cuts of any size.

Provides bounds and applications for crossing number in graphs and surfaces.

Abstract

We prove that if $G$ is a graph with an minimal edge cut $F$ of size three and $G_1$, $G_2$ are the two (augmented) components of $G-F$, then the crossing number of $G$ is equal to the sum of crossing numbers of $G_1$ and $G_2$. Combining with known results, this implies that crossing number is additive over edge-cuts of size $d$ for $d\in\{0, 1, 2, 3\}$, whereas there are counterexamples for every $d\ge 4$. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.