The geodetic number of the lexicographic product of graphs

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