Distance-Edge-Monitoring Sets in Hierarchical and Corona Graphs

Abstract

Let [Formula: see text] and [Formula: see text] be the vertex set and edge set of graph [Formula: see text]. Let [Formula: see text] be the distance between vertices [Formula: see text] and [Formula: see text] in the graph [Formula: see text] and [Formula: see text] be the graph obtained by deleting edge [Formula: see text] from [Formula: see text]. For a vertex set [Formula: see text] and an edge [Formula: see text], let [Formula: see text] be the set of pairs [Formula: see text] with a vertex [Formula: see text] and a vertex [Formula: see text] such that [Formula: see text]. A vertex set [Formula: see text] is distance-edge-monitoring set, introduced by Foucaud, Kao, Klasing, Miller, and Ryan, if every edge [Formula: see text] is monitored by some vertex of [Formula: see text], that is, the set [Formula: see text] is nonempty. In this paper, we determine the smallest size of distance-edge-monitoring sets of hierarchical and corona graphs.