Strong immersions and maximum degree

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(∆(H))|V (H)| whose branch vertices are ∆(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. In this paper, graphs are allowed to have parallel edges and loops, where each loop contributes 2 to the degree of the incident vertex. A graph without parallel edges and loops is called simple. Various containment relations have been studied in structural graph theory. The best known ones are minors and topological minors. A graph H is a minor of G if it can be obtained from G by a sequence of edge and vertex removals and edge contractions. A graph H is a topological minor of G if a subdivision of H is a subgraph of G, or equivalently, if H can be obtained from G by a sequence of edge and vertex removals and by suppressions of vertices of degree two. In a fundamental series of papers, Robertson and Seymour developed the theory of graphs avoiding a fixed minor, giving a description of their structure [14] and proving that every proper minor-closed class of graphs is characterized by a finite set of forbidden minors [15]. The topological minor relation is somewhat harder to deal with (and in particular, there exist proper topological minor-closed classes that are not characterized by a finite set of forbidden topological minors), but a description of their structure is also available [7, 5]. In this paper, we consider a related notion of a graph immersion. Let H and G be graphs. An immersion of H in G is a function θ from vertices and edges of H such that • θ(v) is a vertex of G for each v ∈ V (H), and θ ↾ V (H) is injective. • θ(e) is a connected subgraph of G for each e ∈ E(H), and if f ∈ E(H) is distinct from e, then θ(e) and θ(f) are edge-disjoint. • If e ∈ E(H) is incident with v ∈ V (H), then θ(v) is a vertex of θ(e), and if e is a loop, then θ(e) contains a cycle passing through θ(v). An immersion θ is strong if it additionally satisfies the following condition: Computer Science Institute of Charles University, Prague, Czech Republic. E-mail: [email protected]. Supported the Center of Excellence – Inst. for Theor. Comp. Sci., Prague (project P202/12/G061 of Czech Science Foundation), and by project LH12095 (New combinatorial algorithms decompositions, parameterization, efficient solutions) of Czech Ministry of Education. Institute of Mathematics and DIMAP, University of Warwick, Coventry, UK. E-mail: [email protected]. Her work leading to this invention has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385.