Finding the Most Vital Node of a Shortest Path

In an undirected, 2-node connected graph $G=(V,E)$ with positive real edge lengths, the distance between any two nodes $r$ and $s$ is the length of a shortest path between $r$ and $s$ in $G$. The removal of a node and its incident edges from $G$ may increase the distance from $r$ to $s$. A {\em most vital node} of a given shortest path from $r$ to $s$ is a node (other than $r$ and $s$) whose removal from $G$ results in the largest increase of the distance from $r$ to $s$. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiency the most. In this paper, we show that this problem can be solved in $O(m + n \log n )$ time and $O(m)$ space, where $m$ and $n$ denote the number of edges and the number of nodes in $G$.