THE SIMILARITY OF METRIC DIMENSION AND LOCAL METRIC DIMENSION OF ROOTED PRODUCT GRAPH

Abstract

Let G be a connected graph with vertex set ( )GV and =W {}()....,,, ⊂ The representation of a vertex ( )GVv ∈ with respect to W is the ordered k-tuple ( ) ( ) ( )( ...,,,,, GVwww 21 ()),, k k 1

wvdwvdWvr =| 21 wvd where ()wvd , 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 ( ).dim G l

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.