Geodetic numbers of tensor product and lexicographic product of graphs

A shortest u - v path between two vertices u and v of a graph G is a u - v geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some u - v geodesic of G. For a nonempty subset S of V(G) , let I(S)=boolean OR u,v is an element of SI[u,v] . If I(S)=V(G) , then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by g(G). In this paper, the exact geodetic numbers of the product graphs TxKm and T degrees K<overline>m are obtained, where T is a tree, K<overline>m denotes the complement of the complete graph Km and, x and degrees denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.