Enumerating all geodesics

TL;DR

This paper introduces a new algorithm for enumerating all shortest paths (geodesics) in a graph, leveraging the distance matrix, and compares its performance with existing methods.

Contribution

A novel algorithm for enumerating all geodesics in a graph, utilizing the distance matrix, and a comparative analysis with standard procedures.

Findings

The new algorithm efficiently enumerates all geodesics.

The distance matrix is crucial for related graph tasks.

Comparison shows performance benefits over standard methods.

Abstract

By "geodesic" we mean any sequence of vertices $(v_1,v_2,...,v_k)$ of a graph $G$ that constitute a shortest path from $v_1$ to $v_k$. We propose a novel, natural algorithm to enumerate all geodesics of $G$, and pit it (using Mathematica) against the standard procedure for the task. The distance matrix $D(G)$ plays a crucial role in this. In fact, part of our article is devoted to survey its many uses in related tasks.