Upper bounds on the [formula omitted]-domination number and the [formula omitted]-Roman domination number

Abstract

Let G be a graph and let k be a positive integer. A subset D of the vertex set of G is a k - dominating set if every vertex not in D has at least k neighbors in D . The k - domination number γ k ( G ) is the minimum cardinality among the k -dominating sets of G . A Roman k - dominating function on G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } such that every vertex u for which f ( u ) = 0 is adjacent to at least k vertices v 1 , v 2 , … , v k with f ( v i ) = 2 for i = 1 , 2 , … , k . The weight of a Roman k -dominating function is the value f ( V ( G ) ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight of a Roman k -dominating function on a graph G is called the Roman k - domination number γ k R ( G ) . In 2007, Rautenbach and Volkmann [D. Rautenbach, L. Volkmann, New bounds on the k-domination number and the k-tuple domination number, Appl. Math. Lett. 20 (2007) 98–102] gave an upper bound for the k -domination number γ k ( G ) . Using again the probabilistic method, we achieve better bounds for this parameter and prove new bounds for the k -Roman domination number γ k R ( G ) . Moreover, we generalize known inequalities for the case k = 1 [V.I. Arnautov, Estimations of the external stability number of a graph by means of the minimal degree of vertices, Prikl. Mat. Programm. 11 (1974) 3–8 (in Russian); C. Payan, Sur le nombre d’absorption d’un graphe simple, Cahiers Centre Études Recherche Opér. 17 (1975) 307–317; E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11–22].