A note on forbidding clique immersions

TL;DR

This paper presents a simplified proof of a structure theorem for graphs and Eulerian digraphs that do not contain certain clique immersions, using the Gomory-Hu theorem instead of complex graph minors theory.

Contribution

It introduces a short, alternative proof for the structure of graphs without $K_t$-immersions and extends it to Eulerian digraphs, simplifying previous approaches.

Findings

A short proof for the structure theorem using Gomory-Hu theorem.

Extension of the structure theorem to Eulerian digraphs.

Simplification of the proof technique compared to previous methods.

Abstract

Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without $K_t$-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without $\vec{K}_t$-immersions, where $\vec{K}_t$ denotes the bidirected complete digraph of order $t$.