Characterization of n-vertex graphs with metric dimension n - 3

Abstract

For an ordered set W = {w1, w2,..., wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v{pipe}W):= (d(v, w1), d(v, w2),..., d(v, wk)) is called the metric representation of v with respect to W, where d(x, y) is the distance between vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we characterize all graphs of order n with metric dimension n - 3.