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On Strong Metric Dimension of Diametrically Ver-tex Uniform Graphs

Cyriac Grigorious,Sudeep Stephen, Albert William

Published in:   Vol. 3 Issue 3 Date of Publication:   December 2014
Page(s):   114-116 Publisher:   Integrated Intelligent Research (IIR)
DOI:   10.20894/IJCOA. SAI : 2014SCIA316F0820

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 generator 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 generator for G is called the strong metric dimension of G. The strong met-ric dimension (metric dimension) problem is to find a min-imum strong metric basis (metric basis) in the graph. In this paper, we solve the strong metric dimension and the metric dimension problems for the circulant graph C (n, �{1, 2 . . . j}), 1 ≤ j ≤ ⌊n/2⌋, n ≥ 3 and for the hyper-cubes. We give a lower bound for the problem in case of diametrically uniform graphs. The class of diametrically uniform graphs includes vertex transitive graphs and hence Cayley graphs.