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

For a set M of vertices and an edge e of a graph G, let $ P_G(M, e) $ P G (M , e) be the set of the pair $ (x, y) $ (x , y) with a vertex x of M and a vertex y of $ V(G) $ V (G) such that $ d_G(x, y)\neq d_{G-e}(x, y) $ d G (x , y) ≠ d G − e (x , y). For a vertex x, let $ EM(x) $ EM (x) be the edge set e such that there exists a vertex v in G with $ (x, v) \in P(\{x\}, e) $ (x , v) ∈ P ({ x } , e). 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 $ v\in M $ v ∈ M , that is, for any $ e \in E(G) $ e ∈ E (G) , we have $ P_G(M, e)\neq \emptyset $ P G (M , e) ≠ ∅. The distance-edge-monitoring number of a graph G, denoted by $ \operatorname {dem}(G) $ dem ⁡ (G) , 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. [ABSTRACT FROM AUTHOR]