A new parameter on resolving sets with a realizable triple

A set S of vertices in a graph G is a resolving set if for every pair of vertices u, v ∈ V (G) there is a vertex x ∈ S such that the distances d(x, v) 6= d(x, u). We define a new parameter res(G), the size of the smallest subset S of V (G) that is not a resolving set but every superset of S resolves G. We also demonstrate that for every triple (a, b, c), a ≤ (b + 1) ≤ c, there is a graph G in which a is the metric dimension of G, b = res(G), and c is the resolving number.