Distance-Edge-Monitoring Sets in Hierarchical and Corona Graphs

Let V (G) and E(G) be the vertex set and edge set of graph G. Let d(G)(x,y) be the distance between vertices x and y in the graph G and G - e be the graph obtained by deleting edge e from G. For a vertex set M subset of V (G) and an edge e is an element of E(G), let P(M,e) be the set of pairs (x,y) with a vertex x & ISIN; M and a vertex y & ISIN; V (G) such that d(G)(x,y)&NOTEQUexpressionL;d(G)-e(x,y). A vertex set M subset of V (G) and an edge e is an element of E(G), let P(M,e) be the set of p V (G) is distance-edge-monitoring set, introduced by Foucaud, Kao, Klasing, Miller, and Ryan, if every edge e & ISIN; E(G) is monitored by some vertex of M, that is, the set P(M,e) is nonempty. In this paper, we determine the smallest size of distance-edge-monitoring sets of hierarchical and corona graphs.