Monitoring-edge-geodetic sets in product networks.

Let G be a graph with vertex set $ V(G) $ V (G) and edge set $ E(G) $ E (G). For any $ e \in E(G) $ e ∈ E (G) and $ u,v\in V(G) $ u , v ∈ V (G) , the edge e is monitored by two vertices u and v in graph G if $ d_G(u, v) \neq d_{G-e}(u, v) $ d G (u , v) ≠ d G − e (u , v). A set M of vertices of G is a monitoring-edge-geodetic set of G if for any edge $ e \in E(G) $ e ∈ E (G) there exists a pair $ u, v\in M $ u , v ∈ M such that e is monitored by u, v. The monitoring-edge-geodetic number $ \operatorname {meg}(G) $ meg ⁡ (G) is the cardinality of the minimum MEG-set in G. In this paper, we obtain the exact values or bounds for the MEG numbers of graph products, including join, corona, cluster, lexicographic products and direct products. [ABSTRACT FROM AUTHOR]