2013/04/02 by Dvorak, Zdenek, Klimosova, Tereza · 1 citation
#05C38 (Primary) 05C40 #05C75 #05C83 (Secondary) #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #G.2.2
paper · doi:10.48550/arxiv.1304.0728
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 K1,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.