A grid theorem for strong immersions of walls

TL;DR

This paper establishes a characterization of graphs containing large walls as strong immersion minors through a specific type of tree-cut decomposition, linking structural graph properties with immersion minors.

Contribution

It introduces a new grid theorem connecting strong immersions of walls with tree-cut decompositions based on adhesion and torso path-likeness.

Findings

Graphs with large walls as strong immersion minors lack small-width tree-cut decompositions.

The paper provides a necessary and sufficient condition for the existence of large walls as strong immersions.

A new structural characterization of graphs related to strong immersion minors is proposed.

Abstract

We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small `width', which is measured in terms of its adhesion and the path-likeness of its torsos.