Total Resolvability in Graphs

A set of vertices W is called a weak total resolving, simply written as WTR-set, if for every pair of distinct vertices x,y in G with x∈V(G)\W and y ∈W, there is a vertex w∈W\{y} such that d(x,w )d(y,w) ≠ . A set of vertices W is called a strong total resolving, written as STR-set, if for every pair of distinct vertices x,y in G, there is a vertex w in W such that d(x,w )d(y,w) ≠ for x, yw ≠ . The cardinality of a minimum WTR-set and a minimum STR-set is called the weak total metric dimension and strong total metric dimension of G, denoted by βwt(G) and βst(G), respectively. In this paper, we introduce total metric dimension of graphs and study its relationship with metric dimension and related parameters. We give some realizable results and the maximum order of a connected graph G in terms of diameter and weak total metric dimension of G has also been investigated. 2010 Mathematics subject classification: 05C12