Some comments on the Slater number

Abstract

Let G be a graph with degree sequence d 1 ≥ . . . ≥ d n . Slater proposed sℓ(G) = min{s : (d 1 + 1) + · · · + (d s + 1) ≥ n} as a lower bound on the domination number γ(G) of G. We show that deciding the equality of γ(G) and sℓ(G) for a given graph G is NP-complete but that one can decide efficiently whether+ 1 sℓ(G). For real numbers α and β with α ≥ max{0, β}, let G(α, β) be the class of non-null graphs G such that every non-null subgraph H of G has at most αn(H) − β many edges. Generalizing a result of Desormeaux, Haynes, and Henning, we show that γ(G) ≤ (2α + 1)sℓ(G) − 2β for every graph G in G(α, β) with α ≤ for every tree T of order n at least 2 with n 1 endvertices.

Further results on packing related parameters in graphs

“…The following inequality was independently proved by Gentner and Rautenbach [9], and Desormeaux and Henning [7].…”

Revisiting $k$-tuple dominating sets with emphasis on small values of $k$

“…Slater [18] showed that the domination number of a graph of order n with non-increasing degree sequence d 1 ≥ • • • ≥ d n can be bound from below by the smallest integer t such that t added to the sum of the first t terms of the above-mentioned sequence is at least n. This parameter was first called the Slater number and denoted by sℓ(G) in [6]. This parameter and its properties have been investigated in [10] and [11].…”

“…Note that for some other domination parameters, lower bounds similar to the double Slater number can be obtained. The reader can consult [1], [7] and [11] for more pieces of information about them.…”