Metric dimension of lexicographic product of some known‎ ‎graphs

‎For an ordered set W=\{w_۱,w_۲,\ldots,w_k\} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W):=(d(v,w_۱),d(v,w_۲),\ldots,d(v,w_k)) 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 investigate the metric dimension of the lexicographic product  of graphs G and H, G[H], for some known graphs.

Keywords:

Lexicographic product , Resolving set , Metric‎ ‎dimension , Basis , Adjacency dimension