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An Algorithm for Monitoring Edge-geodetic Sets in Chordal Graphs

Nacim Oijid, Clara Marcille

TL;DR

This work resolves the open question of whether chordal graphs admit a unique minimum meg-set by proving that the set of mandatory vertices $Mand(G)$ is a meg-set for chordal $G$, thereby enabling polynomial-time computation of a minimum meg-set in this class. The proof uses a contradiction based on a minimal counterexample and an elimination step with a simplicial vertex to propagate monitoring properties from $G'$ to $G$, accompanied by structural lemmas about mandatory vertices and their supports. An explicit $O(|V|(|V|+\Delta^2))$ algorithm computes $Mand(G)$, which in chordal graphs yields the minimum meg-set in the same bound. The results settle an open question and suggest future work on extending meg-minimality to more graph classes and exploring density-related and tree-independence perspectives in MEG-set problems.

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 minimal meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimal 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 minimum 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.

An Algorithm for Monitoring Edge-geodetic Sets in Chordal Graphs

TL;DR

This work resolves the open question of whether chordal graphs admit a unique minimum meg-set by proving that the set of mandatory vertices is a meg-set for chordal , thereby enabling polynomial-time computation of a minimum meg-set in this class. The proof uses a contradiction based on a minimal counterexample and an elimination step with a simplicial vertex to propagate monitoring properties from to , accompanied by structural lemmas about mandatory vertices and their supports. An explicit algorithm computes , which in chordal graphs yields the minimum meg-set in the same bound. The results settle an open question and suggest future work on extending meg-minimality to more graph classes and exploring density-related and tree-independence perspectives in MEG-set problems.

Abstract

A monitoring edge-geodetic set (or meg-set for short) of a graph is a set of vertices such that if any edge is removed, then the distance between some two vertices of increases. This notion was introduced by Foucaud et al. in 2023 as a way to monitor networks for communication failures. As computing a minimal meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimal 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 minimum 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.
Paper Structure (8 sections, 14 theorems, 1 figure)

This paper contains 8 sections, 14 theorems, 1 figure.

Table of Contents

  1. Introduction and notations
  2. Some general lemmas
  3. Proof of the main theorem
  4. Overview of the proof
  5. Preliminaries
  6. Proof of the main result
  7. An efficient algorithm to compute $Mand(G)$
  8. Conclusion

Key Result

theorem thmcountertheorem

Let $G$ be a chordal graph. Then $Mand(G)$ is a meg-set of $G$.

Figures (1)

  • Figure 1: A representation of the inclusion relations between the classes of graphs mentioned in the introduction. All graph classes previously known to be meg-minimal are shown in green, chordal graphs are shown in blue, and the graph classes that are known not to be meg-minimal are in orange.

Theorems & Definitions (22)

  • theorem thmcountertheorem
  • theorem thmcountertheorem: Foucaud et al. foucaud:hal-04494089
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma: Foucaud et al. foucaud2023monitoring
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 12 more