The similarity of metric dimension and local metric dimension of rooted product graph

Let G be a connected graph with vertex set V(G) and W = {w1, w2,..., wk}⊂V(G). The representation of a vertex v∈V(G) with respect to W is the ordered k-tuple r(v |W) = (d(v, w1), d(v, w2),..., d(v, wk )), where d(v, w) represents the distance between vertices v and w. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called basis for G. The metric dimension of G, denoted by dim(G), is the number of vertices in a basis of G. If every two adjacent vertices of G have a distinct representation with respect to W, then the set W is called a local resolving set for G and the minimum local resolving set is called a local basis of G. The cardinality of a local basis of G is called local metric dimension of G, denoted by diml (G). In this paper, we study the local metric dimension of rooted product graph and the similarity of metric dimension and local metric dimension of rooted product graph.