Excluding sums of Kuratowski graphs

TL;DR

This paper characterizes graphs that exclude certain sums of Kuratowski graphs as minors, showing they are precisely those with bounded genus, thus linking minor exclusion to topological properties.

Contribution

It provides a complete characterization of graphs excluding sums of Kuratowski graphs as minors in terms of bounded genus.

Findings

Graphs excluding sums of Kuratowski graphs as minors have bounded genus.

The characterization applies to sums of up to three copies of $K_5$ or $K_{3,3}$.

The result bridges minor theory and topological graph properties.

Abstract

We prove that a graph does not contain as a minor a graph formed by 0-, 1-, 2- or 3-summing $k$ copies of $K_5$ or $K_{3,3}$, if and only if it has bounded genus.