An Algorithm for Monitoring Edge-geodetic Sets in Chordal Graphs

TL;DR

This paper proves that chordal graphs have a unique minimal monitoring edge-geodetic set, addressing an open question and contributing to efficient network failure detection methods.

Contribution

It establishes that chordal graphs admit a unique minimal meg-set, enabling polynomial-time computation and advancing understanding of network monitoring in this class.

Findings

Chordal graphs admit a unique minimal meg-set.

This result answers an open question in the field.

It facilitates efficient algorithms for network monitoring.

Abstract

A monitoring edge-geodetic set (or meg-set for short) of a graph is a set of vertices $M$ such that if any edge is removed, then the distance between some two vertices of $M$ increases. This notion was introduced by Foucaud et al. in 2023 as a way to monitor networks for communication failures. As computing a minimum meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimum meg-sets when the input belongs to a restricted class of graphs. Most of these results are based on the property of some classes of graphs to admit a unique minimal meg-set, which is then easy to compute. In this work, we prove that chordal graphs also admit a unique minimal meg-set, answering a standing open question of Foucaud et al.