The metric dimension of the lexicographic product of graphs

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 lexicographic product between two graphs. The lexicographic product of graphs G and H , which is 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 ) is adjacent to ( b , w ) whenever a b ∈ E ( G ) , or a = b and v w ∈ E ( H ) . We give the general bounds of the metric dimension of a lexicographic product of any connected graph G and an arbitrary graph H . We also show that the bounds are sharp.