Local edge and local node connectivity in regular graphs

Abstract

W. Mader [5] proved that every undirected graph (multiple edges are allowed but loops not) contains adjacent nodes x and y joined by min (d(x),dG(y))G edge-disjoint paths and in every undirected simple graph there are two adjacent nodes x and y joined by min (d(x),dG(y) Ginternally node-disjoint paths. In general it is not possible to fix x (or y) arbitrarily. The purpose of this paper is to provide conditions for the existence of a node x in d-regular graphs such that for all y joined to x there are d pairwise edge-disjoint (node-disjoint) paths between x and y. We also examine the directed version in case of local edge connectivity.