A dominating set of a graph G is a subset D of vertices such that every vertex not in D is adjacent to at least one vertex in D. A dominating set D is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in D is paired with exactly one other vertex in D that is within distance 2 from it. The paired domination number, denoted by γ pr (G), is the minimum cardinality of a paired dominating set of G, and the semipaired domination number, denoted by γ pr2 (G), is the minimum cardinality of a semipaired dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ pr (G) ≤ 2 n 4 for any neartriangulation G of order n ≥ 4, and that with some exceptions, γ pr2 (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5.
A subset S of vertices of a connected graph G is a distance-equalizer set if for every two distinct vertices x, y ∈ V (G)\S there is a vertex w ∈ S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related with 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.
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