Computing the Inverse Geodesic Length in Planar Graphs and Graphs of Bounded Treewidth

Abstract

The inverse geodesic length of a graph G is the sum of the inverse of the distances between all pairs of distinct vertices of G . In some domains, it is known as the Harary index or the global efficiency of the graph. We show that, if G is planar and has n vertices, then the inverse geodesic length of G can be computed in roughly O ( n 9/5 ) time. We also show that, if G has n vertices and treewidth at most k , then the inverse geodesic length of G can be computed in O ( n log O ( k ) n ) time. In both cases, we use techniques developed for computing the sum of the distances, which does not have “inverse” component, together with batched evaluations of rational functions.