On the geodetic and the hull numbers in strong product graphs

A set SS of vertices of a connected graph GG is convex, if for any pair of vertices u,v∈Su,v∈S, every shortest path joining uu and vv is contained in SS. The convex hull CH(S)CH(S) of a set of vertices SS is defined as the smallest convex set in GG containing SS. The set SS is geodetic, if every vertex of GG lies on some shortest path joining two vertices in SS, and it is said to be a hull set if its convex hull is V(G)V(G). The geodetic and the hull numbers of GG are the minimum cardinality of a geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also establish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.