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Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, { }-resolving sets were recently introduced. In this paper, we present new results regarding the { }-resolving sets of a graph. In addition to proving general results, we consider {2}-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept ofsolid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for-solid-resolving sets and show how-solid-and { }-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the-solid-and { }-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, {ℓ}-resolving sets were recently introduced. In this paper, we present new results regarding the {ℓ}-resolving sets of a graph. In addition to proving general results, we consider {2}-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept of ℓ-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for ℓ-solid-resolving sets and show how ℓ-solid-and {ℓ}-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the ℓ-solid-and {ℓ}-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

A set R ⊆ V (G) is a resolving set of a graph G if for all distinct vertices v, u ∈ V (G) there exists an element r ∈ R such that d(r, v) = d(r, u). The metric dimension dim(G) of the graph G is the minimum cardinality of a resolving set of G. A resolving set with cardinality dim(G) is called a metric basis of G. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.

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