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.