Fault-tolerance in distance-edge-monitoring sets

Let G be a connected graph. For an edge \(e=xy \in E(G)\), e is monitored by a vertex v if \(d_G(v, y)\ne d_{G-e}(v, y)\) or \(d_G(v, x)\ne d_{G-e}(v, x)\). A set M of vertices of a graph G is distance-edge-monitoring (DEM for short) set if every edge e of G is monitored by some vertex of M. A DEM set X for a graph G is called fault-tolerant DEM set if \(X\setminus \{v\}\) is also DEM set for each v in X. Denote \(\operatorname {dem}(G)\) and \(\operatorname {Fdem}(G)\) the smallest size of DEM set and fault-tolerant DEM sets, respectively. In this paper, we first study the relation between \(\operatorname {Fdem}(G)\) and \(\operatorname {dem}(G)\) for a graph G. Next, we show that \(2 \le \operatorname {Fdem}(G) \le n\) for any graph G with order n. Furthermore, the extremal graphs attaining lower and upper bounds are characterized. In the end, the exact values for some networks are given. Furthermore, it is shown that for \(2\le s<t\le n\), there exists a graph G of order n such that \(\operatorname {dem}(G)=s\) and \(\operatorname {Fdem}(G)=t\).