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Let G be a connected graph with vertex set V , where the distance between two vertices is the length of a shortest path between them. A set S ⊆ V is [ 1 , 2 ] -resolving if each vertex of G is at most distance-two away from a vertex in S and, given a pair of distinct vertices not in S , either there is a vertex in S adjacent to exactly one member of the given pair, or there are two vertices in S each of which is distance-two from exactly one member of the given pair. The [ 1 , 2 ] -dimension of G is the minimum cardinality of a [ 1 , 2 ] -resolving set of G. In this paper, we study the [ 1 , 2 ] -dimension of graphs by proving that the [ 1 , 2 ] -dimension problem is an NP-complete problem, and determine the [ 1 , 2 ] -dimension of some classes of graphs, such as paths, cycles, and full k -ary trees. We also introduce a generalization of metric dimension of which the (original) metric dimension and the [ 1 , 2 ] -dimension, as well as other metric dimension variants in the literature, are special instances. [ABSTRACT FROM AUTHOR]

Let G be a connected graph, and let D (G) be the set of all dominating (multi)sets for G. For D 1 and D 2 in D (G) , we say that D 1 is single-step transferable to D 2 if there exist u ∈ D 1 and v ∈ D 2 , such that u v ∈ E (G) and D 1 − { u } = D 2 − { v }. We write D 1 ⟶ ∗ D 2 if D 1 can be transferred to D 2 through a sequence of single-step transfers. We say that G is k -transferable if D 1 ⟶ ∗ D 2 for any D 1 , D 2 ∈ D (G) with | D 1 | = | D 2 | = k. The transferable domination number of G is the smallest integer k to guarantee that G is l -transferable for all l ≥ k. We study the transferable domination number of graphs in this paper. We give upper bounds for the transferable domination number of general graphs and bipartite graphs, and give a lower bound for the transferable domination number of grids. We also determine the transferable domination number of P m × P n for the cases that m = 2 , 3 , or m n ≡ 0 (mod 6). Besides these, we give an example to show that the gap between the transferable domination number of a graph G and the smallest number k so that G is k -transferable can be arbitrarily large. [ABSTRACT FROM AUTHOR]