The annihilation number does not bound the 2-domination number from the above

Abstract

The 2-domination number γ 2 (G) of a graph G is the minimum cardinality of a set S ⊆ V (G) such that every vertex from V (G) \ S is adjacent to at least two vertices in S. 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 its edges. It was conjectured that γ 2 (G) ≤ a(G) + 1 holds for every connected graph G. The conjecture was earlier confirmed, in particular, for graphs of minimum degree 3, for trees, and for block graphs. In this paper, we disprove the conjecture by proving that the 2-domination number can be arbitrarily larger than the annihilation number. On the positive side we prove the conjectured bound for a large subclass of bipartite, connected cacti, thus generalizing a result of

On the equality of domination number and 2-domination number

“…The concept of k-domination in graphs was introduced by Fink and Jacobson [15,16] and it has been studied extensively by many researchers (see for example [5][6][7][8]10,13,14,18,19,26,30,32]). For more details, we refer the reader to the books on domination by Haynes, Hedetniemi and Slater [23,24] and to the survey on k-domination and k-independence by Chellali et al [9].…”

Desigualdades Do Tipo Nordhaus-Gaddum Para O Número De Aniquilação

“…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.…”

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

“…We add that a conjecture parallel to 1.1 has been posed also for the 2-domination number of a graph G. In [8] (see also [13]), the latter conjecture was verified for trees, and in [11] for block graphs. In [23], the conjecture was disproved by demonstrating that the 2-domination number can be arbitrarily larger than the annihilation number. However, the counterexamples presented are far from being counterexamples for Conjecture 1.1 and the authors say that they are "inclined to believe that Conjecture 1.1 holds true."…”