Title: Max flow vitality in general and $st$-planar graphs,
Abstract: The \emph{vitality} of an arc/node of a graph with respect to the maximum
flow between two fixed nodes $s$ and $t$ is defined as the reduction of the
maximum flow caused by the removal of that arc/node. In this paper we address
the issue of determining the vitality of arcs and/or nodes for the maximum flow
problem. We show how to compute the vitality of all arcs in a general
undirected graph by solving only $2(n-1)$ max flow instances and, In
$st$-planar graphs (directed or undirected) we show how to compute the vitality
of all arcs and all nodes in $O(n)$ worst-case time. Moreover, after
determining the vitality of arcs and/or nodes, and given a planar embedding of
the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in
time proportional to the size of the set.