BRIDGES IN HIGHLY CONNECTED GRAPHS

Let P - {P(1),..., P(l)} be a set of internally disjoint paths contained in a graph G, and let S be the subgraph defined by U(i=1)(t) P(i). A P-bridge is either an edge of G - E(S) with both endpoints in V (S) or a component C of G - V (S) along with all the edges from V (C) to V (S). The attachments of a bridge B are the vertices of V (B) boolean AND V (S). A bridge B is k-stable if there does not exist a subset of at most k - 1 paths in P containing every attachment of B. A classic theorem of Tutte [Graph Theory, Addison-Wesley, Menlo Park, CA, 1984] states that if G is a 3-connected graph, there exists a set of internally disjoint paths P' = {P'(1),..., P'(l)} such that P(i) and P'(i) have the same endpoints for 1 <= i <= t and every P'-bridge is 2-stable. We prove that if the graph is sufficiently connected, the paths P'(1),..., P'(l) may be chosen so that every bridge containing at least two edges is, in fact, k-stable. We also give several simple applications of this theorem related to a conjecture of Lovasz [Problems in Graph Theory, Recent Advances in Graph Theory, M. Felder, ed., Acadamia, Prague, 1975] on deleting paths while maintaining high connectivity.