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

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.