An Algorithm for Monitoring Edge-geodetic Sets in Chordal Graphs

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.

💡 Research Summary

The paper addresses the problem of monitoring edge‑geodetic sets (MEG‑sets), a concept introduced by Foucaud et al. (2023) for detecting communication failures in network models represented by graphs. A set M of vertices is a MEG‑set if, for every edge e, there exist two vertices a, b ∈ M such that every shortest a‑b path contains e; consequently, the removal of e increases the distance between a and b, signalling a failure. While finding a minimum‑size MEG‑set is NP‑hard in general, several restricted graph families (interval graphs, cographs, block graphs, strongly chordal graphs, etc.) have been shown to be “meg‑minimal”, meaning they possess a unique minimum MEG‑set that coincides with the set of mandatory vertices (those that belong to every MEG‑set).

The open question left by previous work was whether the larger class of chordal graphs is also meg‑minimal. This paper resolves the question affirmatively. The authors proceed by contradiction: assume a minimal counter‑example G (a chordal graph that is not meg‑minimal) with the smallest possible number of vertices. Because chordal graphs always contain a simplicial vertex (a vertex whose neighbours form a clique), they select such a vertex v and consider the induced subgraph G′ = G − {v}. By minimality, G′ must be meg‑minimal, so its mandatory set M′ is its unique minimum MEG‑set.

The authors define W ⊆ N(v) as the vertices that are mandatory in G′ but not in G, and construct the candidate set M = {v} ∪ (M′ \ W). Lemma 1 (derived from a known characterization of mandatory vertices) guarantees that every simplicial vertex is mandatory, so v ∈ M. Lemma 5 shows that any vertex outside N(v) that is mandatory in G′ remains mandatory in G, while Lemma 6 proves that vertices in M′ \ W stay mandatory after adding v. Consequently, M consists solely of mandatory vertices of G.

Lemma 7 establishes that a minimal counter‑example must be 2‑vertex‑connected; otherwise, a cut‑vertex would allow the construction of a MEG‑set for G from the MEG‑sets of its components, contradicting minimality.

The core of the proof is a case analysis showing that every edge e of G is monitored by two vertices of M. If e lies entirely in G′ and is already monitored by a pair from M′ \ W, Lemma 4 ensures the same pair monitors e in G. The more delicate situation occurs when one of the monitoring vertices belongs to W. The authors distinguish whether the two vertices are adjacent. If they are non‑adjacent, they examine shortest paths from v to the non‑W vertex and use the chordal property (Lemma 2) to locate a chord that forces the existence of another vertex y that, together with the non‑W vertex, monitors e. If the vertices are adjacent, they either find that v together with the non‑W vertex monitors e, or they identify a neighbor a₁ of the W‑vertex that participates in a 4‑cycle, again yielding a suitable monitoring pair.

Lemma 8 is employed to argue that whenever a pair {α,β} monitors an edge, all shortest α‑a paths or all shortest β‑b paths must contain the edge, which is crucial for the adjacency‑based arguments.

Finally, Lemma 9 confirms that the constructed set M is indeed a MEG‑set of G. Since M contains only mandatory vertices, it must be the unique minimum MEG‑set of G. This contradicts the assumption that G was a counter‑example, proving that every chordal graph is meg‑minimal.

As a direct consequence, the known polynomial‑time algorithm for extracting mandatory vertices (based on checking the existence of a supporting vertex for each candidate) yields a polynomial‑time algorithm for computing a minimum MEG‑set on any chordal graph. The result bridges a gap between theoretical graph classes and practical network monitoring, offering an efficient method to place a minimal set of probes that can detect any single‑edge failure via distance changes.