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 \emph{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 Sierpi{\'n}ski-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 Sierpi{\'n}ski-type graphs.