Excluding Graphs as Immersions in Surface Embedded Graphs

TL;DR

This paper characterizes graphs embedded in surfaces that exclude a fixed graph as an immersion, showing they have either bounded treewidth or small edge connectivity, and proves an excluded grid theorem for such graphs.

Contribution

It provides a structural characterization of surface-embedded graphs excluding a fixed immersion and establishes an excluded grid theorem for bounded genus graphs.

Findings

Graphs excluding a fixed immersion have bounded treewidth or small edge connectivity.

A new excluded grid theorem for bounded genus graphs is proved.

Structural properties depend on the surface's Euler genus and the excluded graph.

Abstract

We prove a structural characterization of graphs that forbid a fixed graph $H$ as an immersion and can be embedded in a surface of Euler genus $\gamma$. In particular, we prove that a graph $G$ that excludes some connected graph $H$ as an immersion and is embedded in a surface of Euler genus $\gamma$ has either "small" treewidth (bounded by a function of $H$ and $\gamma$) or "small" edge connectivity (bounded by the maximum degree of $H$). Using the same techniques we also prove an excluded grid theorem on bounded genus graphs for the immersion relation.