Immersions in highly edge connected graphs

TL;DR

This paper investigates the edge connectivity conditions needed for graphs to contain a fixed graph as an immersion, providing structural characterizations for both weak and strong immersions in highly edge-connected graphs.

Contribution

It introduces a tree-cut decomposition framework for weak immersions and a structure theorem for highly edge-connected graphs avoiding fixed cliques as strong immersions.

Findings

Graphs with maximum degree D and high edge connectivity have bounded tree-cut width if they do not contain a fixed graph as a weak immersion.

Highly edge-connected graphs can lack a fixed clique as a strong immersion, characterized by a new structure theorem.

Abstract

We consider the problem of how much edge connectivity is necessary to force a graph G to contain a fixed graph H as an immersion. We show that if the maximum degree in H is D, then all the examples of D-edge connected graphs which do not contain H as a weak immersion must have a tree-like decomposition called a tree-cut decomposition of bounded width. If we consider strong immersions, then it is easy to see that there are arbitrarily highly edge connected graphs which do not contain a fixed clique K_t as a strong immersion. We give a structure theorem which roughly characterizes those highly edge connected graphs which do not contain K_t as a strong immersion.