A structure theorem for strong immersions

TL;DR

This paper characterizes the structure of graphs that do not contain a fixed graph as a strong immersion, providing insights into how such graphs are constructed and constrained.

Contribution

It introduces a structure theorem for graphs avoiding a fixed graph as a strong immersion, advancing understanding of graph containment relations.

Findings

Characterization of graphs avoiding a fixed strong immersion

Structural decomposition results for such graphs

Implications for graph minor and immersion theory

Abstract

A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge-disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion.