On the metric dimension of circulant and Harary graphs (original) (raw)

A metric generator is a set W of vertices of a graph GðV; EÞ such that for every pair of vertices u; v of G, there exists a vertex w 2 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. In this case the vertex w is said to resolve or distinguish the vertices u and v. The minimum cardinality of a metric generator for G is called the metric dimension. The metric dimension problem is to find a minimum metric generator in a graph G. In this paper, we make a significant advance on the metric dimension problem for circulant graphs Cðn; AEf1; 2; . . . ; jgÞ; 1 6 j 6 bn=2c; n P 3, and for Harary graphs.