Graphs with distinguishing sets of size k

The size of a resolving set R of a non-trivial connected graph Gamma of order n >= 2 is the number of edges in the induced subgraph <R>. The minimum cardinality of a resolving set of size k of graph Gamma is called the metric dimension of size k, denoted by beta((k))(Gamma). We study the existence of resolving sets of size k in some families of graphs and investigate their properties. We find bounds on the metric dimension of size k of a graph Gamma. We give the necessary condition for the metric dimension of size k and size (k + 1) of a graph Gamma, to satisfy the inequality beta((k+1))(Gamma) - beta((k))(Gamma) <= 1. We will disprove a conjecture on bounds of the metric dimension of size k. For every positive integers k, l, and n such that k + 1 <= l <= n, we give a realizable result of a graph Gamma of order n and l = beta((k))(Gamma).