Distance-edge-monitoring numbers of some related pseudo wheel networks

Abstract

For a set M of vertices and an edge e of a graph G, let be the set of the pair with a vertex x of M and a vertex y of such that . For a vertex x, let be the edge set e such that there exists a vertex v in G with . A set M of vertices of a graph G is distance-edge-monitoring set if every edge e of G is monitored by some vertex , that is, for any , we have . The distance-edge-monitoring number of a graph G, denoted by , is defined as the smallest size of distance-edge-monitoring sets of G. In this paper, we study the distance edge monitoring number of pseudo wheel graphs, that is, some variants of wheel graph.