Rooted minors and locally spanning subgraphs

Abstract

AbstractResults on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider “local” versions of such statements. In 1966, for instance, D. W. Barnette proved that a 3‐connected planar graph contains a spanning tree of maximum degree at most 3. A local translation of this statement is that if is a planar graph, is a subset of specified vertices of such that cannot be separated in by removing two or fewer vertices of , then has a tree of maximum degree at most 3 containing all vertices of . Our results constitute a general machinery for strengthening statements about ‐connected graphs (for ) to locally spanning versions, that is, subgraphs containing a set of a (not necessarily planar) graph in which only has high connectedness. Given a graph and , we say is a minor of rooted at , if is a minor of such that each bag of contains at most one vertex of and is a subset of the union of all bags. We show that has a highly connected minor rooted at if cannot be separated in by removing a few vertices of . Combining these investigations and the theory of Tutte paths in the planar case yields locally spanning versions of six well‐known results about degree‐bounded trees, Hamiltonian paths and cycles, and 2‐connected subgraphs of graphs.