# Computing the inverse geodesic length in planar graphs and graphs of bounded treewidth

## TL;DR

This paper presents efficient algorithms for computing the inverse geodesic length, also known as the Harary index, in planar graphs and graphs with bounded treewidth, improving computational complexity for these classes.

## Contribution

It introduces algorithms with sub-quadratic and near-linear time complexities for calculating the inverse geodesic length in specific graph classes, extending existing distance sum techniques.

## Key findings

- Inverse geodesic length can be computed in roughly O(n^{9/5}) time for planar graphs.
- In graphs with bounded treewidth, it can be computed in O(n log^{O(k)} n) time.
- Techniques from distance sum computations and rational function evaluations are adapted for the inverse case.

## 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.