Graph Odometry

TL;DR

This paper investigates how to determine all edge weights in a graph using non-backtracking closed walks from a single vertex, establishing conditions for exact recovery and the minimum number of walks needed.

Contribution

It provides necessary and sufficient conditions for edge weight determination from a single vertex and quantifies the minimum number of walks required.

Findings

Edge weights can be exactly determined if the graph has minimum degree at least three.

The minimum number of walks needed for complete edge weight recovery is characterized.

The method applies to non-backtracking closed walks from any starting vertex.

Abstract

We address problem of determining edge weights on a graph using non-backtracking closed walks from a vertex. We show that the weights of all of the edges can be determined from any starting vertex exactly when the graph has minimum degree at least three. We also determine the minimum number of walks required to reveal all edge weights.