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,K-d(Delta(H))vertical bar V(H)vertical bar 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 G of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if G has either unbounded maximum degree or unbounded tree-width.