The metric dimension of the lexicographic product of graphs

For a set W of vertices and a vertex v in a connected graph G , the k -vector r W ( v ) = ( d ( v , w 1 ) , … , d ( v , w k ) ) is the metric representation of v with respect to W , where W = { w 1 , … , w k } and d ( x , y ) is the distance between the vertices x and y . The set W is a resolving set for G if distinct vertices of G have distinct metric representations with respect to W . The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we study the metric dimension of the lexicographic product of graphs G and H , denoted by G [ H ] . First, we introduce a new parameter, the adjacency dimension, of a graph. Then we obtain the metric dimension of G [ H ] in terms of the order of G and the adjacency dimension of H .