Max Flow Vitality of Edges and Vertices in Undirected Planar Graphs

We study the problem of computing the vitality with respect to max flow of edges and vertices in undirected planar graphs, where the vitality of an edge/vertex in a graph with respect to max flow between two fixed vertices $s,t$ is defined as the max flow decrease when the edge/vertex is removed from the graph. We show that the vitality of any $k$ selected edges can be computed in $O(kn + n\log\log n)$ worst-case time, and that a $\delta$ additive approximation of the vitality of all edges with capacity at most $c$ can be computed in $O(\frac{c}{\delta}n +n \log \log n)$ worst-case time, where $n$ is the size of the graph. Similar results are given for the vitality of vertices. All our algorithms work in $O(n)$ space.