Characterizing optimal monitoring edge-geodetic sets for some structured graph classes

TL;DR

This work advances the study of the MIN-MEG problem by providing polynomial-time algorithms for computing meg(G) on several structured graph classes. It centers on the structure of mandatory vertices and its relation to cut-vertices, establishing exact characterizations such as Man(G)=V\setminus Cut(G) for distance-hereditary and bipartite permutation graphs, and meg(G)=|Man(G)| for $P_4$-sparse and strongly chordal graphs. The proofs combine one-vertex-extension orderings, strong elimination orderings, and spider-based decompositions, yielding constructive polynomial-time solutions. These results extend prior tractable cases (e.g., interval graphs, cographs) and deepen understanding of when MEG can be efficiently computed, with implications for network monitoring tasks.

Abstract

Given a graph $G=(V,E)$, a set $S\subseteq V$ is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in $S$. The minimum size of such a set in $G$ is called the monitoring edge-geodetic number of $G$ and is denoted by $meg(G)$. In this work, we compute the monitoring edge-geodetic number efficiently for the following graph classes: distance-hereditary graphs, $P_4$-sparse graphs, bipartite permutation graphs, and strongly chordal graphs. The algorithms follow from structural characterizations of the optimal monitoring edge-geodetic sets for these graph classes in terms of \emph{mandatory vertices} (those that need to be in every solution). This extends previous results from the literature for cographs, interval graphs and block graphs.

Figures (12)

• Figure 1: Inclusion diagram for graph classes mentioned in this paper (and related ones). If a class $A$ has an upward path to class $B$, then $A$ is included in $B$. For any graph $G$ in a dark gray class, we have $\vert Man(G)\vert=\vert V(G)\setminus Cut(G)\vert=meg(G)$. For any graph $G$ in a light gray class, we have $\vert Man(G)\vert=meg(G)$. Results for boxes with a thick border are proved in this paper.
• Figure 2: Examples of spiders with spider partition $(S,C,R)$
• Figure 3: The mandatory vertices in $V(G_k)\setminus Cut(G_k)$ remain mandatory in $V(G)\setminus Cut(G)$
• Figure 4: Figures used in the proof of Lemma \ref{['lem:bip_per']}
• Figure 5: $v$ is a mandatory in $G_{k+1}$ but not in $G_k$

Theorems & Definitions (36)

• Theorem 2.1: DBLP:conf/caldam/FoucaudMMSST24
• Theorem 2.2: DBLP:journals/corr/abs-2210.03774DBLP:conf/caldam/FoucaudNS23
• Theorem 2.3: DBLP:conf/caldam/FoucaudMMSST24
• Theorem 2.4: DBLP:journals/corr/abs-2210.03774DBLP:conf/caldam/FoucaudNS23
• Theorem 3.1
• proof
• Theorem 4.1: Defn_P4sparse
• Lemma 4.1
• proof
• Lemma 4.2