Strong immersions and maximum degree

TL;DR

This paper investigates the conditions under which a graph H can be strongly immersed in another graph G, establishing a function that guarantees the immersion based on the presence of a large star subgraph with certain connectivity properties.

Contribution

The authors introduce a new function relating maximum degree and size of star immersions, providing structural insights into graphs avoiding certain strong immersions.

Findings

Existence of a function d:N->N linking star immersions to strong immersions of H.

Characterization of classes of graphs containing all degree-4 graphs as strong immersions.

Structural consequences for graphs avoiding strong immersions of a fixed graph H.

Abstract

A graph H is strongly immersed in G if G is obtained from H 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 show that there exists a function d:N->N such that for all graphs H and G, if G contains a strong immersion of the star K_{1,d(Delta(H))|V(H)|} whose branch vertices are Delta(H)-edge-connected to one another, then H is strongly immersed in G. This has a number of structural consequences for graphs avoiding a strong immersion of H. In particular, a class C of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if C has either unbounded maximum degree or unbounded tree-width.