  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.
