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 3g(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)=3g(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.

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