Bridges in Highly Connected Graphs

Abstract

Let $\mathcal{P}=\{P_1,\dots,P_l\}$ be a set of internally disjoint paths contained in a graph G, and let S be the subgraph defined by $\bigcup_{i=1}^{t}P_i$. A $\mathcal{P}$-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 $V(B)\cap V(S)$. A bridge B is k-stable if there does not exist a subset of at most $k-1$ paths in $\mathcal{P}$ 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 $\mathcal{P}'=\{P_1',\dots,P_l'\}$ such that $P_i$ and $P_i'$ have the same endpoints for $1\leq i\leq t$ and every $\mathcal{P}'$-bridge is 2-stable. We prove that if the graph is sufficiently connected, the paths $P_1',\dots,P_l'$ 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 Lovász [Problems in Graph Theory, Recent Advances in Graph Theory, M. Felder, ed., Acadamia, Prague, 1975] on deleting paths while maintaining high connectivity.