Strong Immersions and Maximum Degree

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 $d\colon N\to N$ 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 $\mathcal{G}$ of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if $\mathcal{G}$ has either unbounded maximum degree or unbounded tree-width.