The structure of group-labeled graphs forbidding an immersion

TL;DR

This paper establishes a structural characterization of group-labeled graphs that exclude a specific immersion, revealing a tree-cut decomposition with constrained properties related to group labels and vertex degrees.

Contribution

It provides a novel structure theorem for $ ext{Gamma}$-labeled graphs forbidding a fixed immersion, extending understanding of their decomposition based on group labels.

Findings

Graphs admit a tree-cut decomposition with specific properties.

Bags contain few high degree vertices or are nearly signed over a subgroup.

The structure theorem applies to any finite group $ ext{Gamma}$.

Abstract

A $\Gamma$-labeled graph is an oriented graph with edges invertibly labeled by a group $\Gamma$. We prove a structure theorem for $\Gamma$-labeled graphs which forbid a fixed $\Gamma$-labeled graph as an immersion, for any finite $\Gamma$. Roughly, we show that such graphs admit a tree-cut decomposition in which every bag either contains few high degree vertices or is nearly signed over a proper subgroup of $\Gamma$.