Total domination in plane triangulations

Abstract

A total dominating set of a graph G = (V, E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total 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 t (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5, with two exceptions.

Paired and semipaired domination in triangulations

“…In [7], the authors prove that γ t (T ) ≤ 2n 5 for any near-triangulation T , with some exceptions. Their proof is based on a double induction and what the authors call reducible and irreducible near-triangulations, and terminal polygons in irreducible near-triangulations.…”

“…We will see in the forthcoming sections that these techniques can be also used to bound the paired and semipaired domination numbers in neartriangulations. Following the definitions and notation in [7], in this section we review those techniques and give some related results.…”

“…In the same paper, the authors conjectured that γ(G) ≤ n 4 for every n-vertex triangulation G, when n is large enough. Since then, several papers have been devoted to either trying to prove that conjecture [23,31,34], or showing thigh bounds, mainly for MOPs, for several variants of the domination number [2,3,4,6,7,9,10,19,26,31,35,37]. In particular, Canales et al [3] proved that γ pr (G) ≤ 2 n 4 for any MOP G of order n ≥ 4, and Henning and Kaemawichanurat [19] showed that γ pr2 (G) ≤ 2 5 n for any MOP G of order n ≥ 5, except for a special family F of MOPs of order 9.…”

“…The proofs of these two results strongly rely on the techniques given in [7] to show that γ t (G) ≤ 2n 5 for near-triangulations, where γ t (G) is the total domination number of G. Given a simple graph G = (V, E), a subset D ⊆ V such that every vertex in V is adjacent to a vertex in D is called a total dominating set (TD-set for short) of G, and that the total domination number, denoted by γ t (G), is the minimum cardinality of a total dominating set of G.…”

“…The paper is organized as follows. Section 2 is devoted to give some terminology and basic results on near-triangulations, and to revise the techniques used in [7] that we need to prove our main results. In Sections 3 and 4, we use these techniques to bound the paired and semipaired domination numbers in near-triangulations, respectively.…”