The geodetic number of lexicographic product of graphs ∗
نویسندگان
چکیده
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 non-complete graphs 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. ∗Work supported in part by the Ministry of Science and Technology of Slovenia under the grants J1-2043 and P1-0297. The authors are also with the Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana.
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