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.