Bounding the paired-domination number of a tree in terms of its annihilation number

Abstract

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a paired-dominating set of G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T of order n ≥ 2, γ pr (T) ≤ 4a(T)+2 3 and we characterize the trees achieving this bound.

Smallest domination number and largest independence number of graphs and forests with given degree sequence

“…Many known bounds on the domination number and the independence number depend only on the degree sequence, or on derived quantities such as the order, the size, the minimum degree, and the maximum degree . For a graph G with nonincreasing degree sequence , Slater observed where and Pepper observed where is known as the annihilation number of G . Clearly, and .…”

An upper bound on the double domination number of trees

“…Clearly, every double dominating set of T can be extended to a double domination set of T by adding the vertices v 1 , v 2 , w and hence γ ×2 (T ) ≤ γ ×2 (T ) + 3. Assume that S = S ∪ {v 1 , w} when v 3 ∈ S and S = (S − {v 3 …”

“…In [15] and [16], Pepper proved that the annihilation number is an upper bound on the independence number of a graph and in [14] the case for equality of the upper bound was characterized by Larson and Pepper. The relation between annihilation number and some graph parameters have been studied by several authors. For instance, DeLaViña et al presented an upper bound on 2-domination number in terms of annihilation number for some classes of graphs [6], Dehgardi et al investigated the relation between some domination parameters and the annihilation number of trees [3,4,5], Desormeaux et al proved that for any tree T , a(T ) + 1 is an upper bound on the total domination number [8].…”

Desigualdades Do Tipo Nordhaus-Gaddum Para O Número De Aniquilação

“…Aouchiche e P. Hansen (2013) organizaram as desigualdades do tipo Nordha-us-Gaddum para diversos invariantes, inclusive para invariantes cujas definições dependem das cardinalidades de subconjuntos específicos do grafo, como o número de independência, o número de dominação, o número romano de dominação, número de dominação total, entre outros. A relação entre esses parâmetros de dominação e o número de aniquilação foi estudada por vários autores (DEHGARDAI; NOROUZIAN;KHOEILAR et al, 2018;DELAVIÑA et al, 2010;DESORMEAUX;HAYNES;HENNING, 2013;NING;Revista Mundi, Engenharia e Gestão, Paranaguá, PR, v. 6, n. 3, p. 360-01, 360-09, 2021. DOI: 10.21575/25254782rmetg2021vol6n31662 LU; WANG, 2019;DEHGARDI;KHODKAR, 2013;DESORMEAUX et al, 2014;YUE et al, 2020), estabelecendo uma valiosa conexão com as desigualdades do tipo Nordhaus-Gaddum.…”