Geodesic transversal problem for join and lexicographic product of graphs

A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. The minimum cardinality of a geodesic transversal of G is denoted by gt(G) and is called geodesic transversal number. For two graphs G and H we deal with the behavior of this invariant for the lexicographic product G omicron H and join G circle plus H. We determine gt(G circle plus H) in terms of structural properties of the original graphs and describe gt(G omicron H) as a solution of an optimization problem concerning specific subsets of V (G).