On the Distance-Edge-Monitoring Numbers of Strong Product Graphs

Let [Formula: see text] be a graph with vertex set [Formula: see text], [Formula: see text] be a subset of [Formula: see text], and [Formula: see text] be an edge in [Formula: see text], and let [Formula: see text] be the set of pairs [Formula: see text] such that [Formula: see text] where [Formula: see text] and [Formula: see text]. For a vertex [Formula: see text], let [Formula: see text] be the set of edges [Formula: see text] such that there exists a vertex [Formula: see text] in [Formula: see text] with [Formula: see text]. [Formula: see text] is called a distance-edge-monitoring set if every edge [Formula: see text] of [Formula: see text] is monitored by some vertex of [Formula: see text], that is, the set [Formula: see text] is nonempty. The distance-edge-monitoring number of [Formula: see text], denoted by [Formula: see text], is defined as the smallest size of distance-edge-monitoring sets of [Formula: see text]. In this paper, for two graphs [Formula: see text] and [Formula: see text], we prove that each edge in [Formula: see text] and [Formula: see text] can only be monitored by the vertices incident with it, where ⊠ is the strong product operation and [Formula: see text]. Moreover, we determine distance-edge-monitoring number of strong product of two specific graphs, including [Formula: see text] and [Formula: see text], where [Formula: see text].