The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S . The size of a minimum geodetic set in G is the geodetic number g ( G ) of G . We find that the geodetic number of the lexicographic product G ∘ H for a non-complete graph H lies between 2 and 3 g ( G ) . We characterize the graphs G and H for which g ( G ∘ H ) = 2 , as well as the lexicographic products T ∘ H that enjoy g ( T ∘ H ) = 3 g ( G ) , when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G , a formula that expresses the exact geodetic number of G ∘ H is established, where G is an arbitrary graph and H a non-complete graph.