THE METRIC DIMENSION OF A GRAPH COMPOSITION PRODUCTS WITH STAR

A set of vertices W resolves a graph G if every vertex is uniquely determined by its coordinate of distances to the vertices in W . The minimum cardinality of a resolving set of G is called the metric dimension of G. In this paper, we consider a graph which is obtained by the composition product between two graphs. The composition product of graphs G and H, denoted by G[H], is the graph with vertex set V (G) × V (H) = {(a, v)|a ∈ V (G); v ∈ V (H)}, where (a, v) adjacent with (b, w) whenever ab ∈ E(G), or a = b and vw ∈ E(H). We give a general bound of the metric dimension of a composition product of any connected graph G and a star. We also show that the bound is sharp.