On the Strong Metric Dimension of Tetrahedral Diamond version 11.dvi

A metric basis is a set W of vertices of a graph G(V,E) such that for every pair of vertices u, v of G, there exists a vertex w ∈ W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. The minimum cardinality of a metric basis for G is called the metric dimension. A pair of vertices u, v is said to be strongly resolved by a vertex s, if there exist at least one shortest path from s to u passing through v, or a shortest path from s to v passing through u. A set W ⊆ V , is said to be a strong metric basis if for all pairs u, v / ∈ W , there exist some element s ∈ W such that s strongly resolves the pair u, v. The smallest cardinality of a strong metric basis for G is called the strong metric dimension of G. The strong metric dimension (metric dimension) problem is to find a minimum strong metric basis (metric basis) in the graph. In this paper, we solve the strong metric dimension and the metric dimension problems for the graph of tetrahedral diamond.