The Metric Dimension of the Composition Product of Graphs

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) := (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the (metric) representation of v with respect to W , where d(x, y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct 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 composition product of graphs G and H, G[H]. First, we introduce a new parameter which is called adjacency metric dimension of a graph. Then, we obtain the metric dimension of G[H] in terms of the order of G and the adjacency metric dimension of H.