Excluding disjoint Kuratowski graphs

TL;DR

This paper investigates graphs that exclude certain disjoint unions of Kuratowski graphs as minors, showing that such graphs can be simplified by removing a bounded number of vertices to become surface-embeddable without these minors.

Contribution

It establishes a structural characterization of graphs excluding disjoint Kuratowski minors, extending the understanding of graph minors and surface embeddings.

Findings

Graphs excluding disjoint Kuratowski minors can be made surface-embeddable after removing a bounded set of vertices.

The result generalizes previous minor exclusion theorems to disjoint unions of Kuratowski graphs.

Provides a method to identify a small vertex set to simplify complex graphs for topological analysis.

Abstract

A graph is a ``$k$-Kuratowski graph'' if it has exactly $k$ components, each isomorphic to $K_5$ or to $K_{3,3}$. We prove that if a graph $G$ contains no $k$-Kuratowski graph as a minor,then there is a set $X$ of boundedly many vertices such that $G\setminus X$ can be drawn in a (possibly disconnected) surface in which no $k$-Kuratowski graph can be drawn.