Distance-edge-monitoring sets of networks

It is important to be able to monitor the network and detect this failure when a connection (an edge) fails. For a vertex set M and an edge e of the graph G, let P(M, e) be the set of pairs (x, y) with a vertex x of M and a vertex y of V(G) such that e belongs to all shortest paths between x and y. A vertex set M of the graph G is distance-edge-monitoring set if every edge e of G is monitored by some vertex of M, that is, the set P(M, e) is nonempty. The distance-edge-monitoring number of a graph G, recently introduced by Foucaud, Kao, Klasing, Miller, and Ryan, is defined as the smallest size of distance-edge-monitoring sets of G. In this paper, we determine the bounds of the distance-edge-monitoring number of grid-based pyramids and the exact value of distance-edge-monitoring number for M(t)-graph and Sierpiński-type graphs. We also compare the distance-edge-monitoring set with average degree, the size of edge set and the size of vertex set of G, where G is M(t)-graph or Sierpiński-type graphs.