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For an ordered set W= w₁, w₂, , wₖ of vertices in a connected graph G and a vertex v of G, the code of v with respect to W is the k-vector \ cW (v) = (d (v, w₁), d (v, w₂), , d (v, wₖ) ). \ The set W is an independent resolving set for G if (1) W is independent in G and (2) distinct vertices have distinct codes with respect to W. The cardinality of a minimum independent resolving set in G is the independent resolving number ir (G). We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs G of order n with ir (G) = 1, n-1, n-2, and present several realization results. It is shown that for every pair r, k of integers with k 2 and 0 r k, there exists a connected graph G with ir (G) = k such that exactly r vertices belong to every minimum independent resolving set of G.
Chartrand et al. (Wed,) studied this question.