Some comments on the Slater number

Let G be a graph with degree sequence d1≥⋯≥dn. Slater proposed sℓ(G)=min{s:(d1+1)+⋯+(ds+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 γ(G)>sℓ(G) or γ(G)≤lnn(G)sℓ(G)+1sℓ(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≤32. Furthermore, we show that γ(G)∕sℓ(G) is bounded for graphs G in G(α,β) if and only if α<2. For an outerplanar graph G with sℓ(G)≥2, we show γ(G)≤6sℓ(G)−6. In analogy to sℓ(G), we propose sℓt(G)=min{s:d1+⋯+ds≥n} as a lower bound on the total domination number. Strengthening results due to Raczek as well as Chellali and Haynes, we show that sℓt(T)≥n+2−n12 for every tree T of order n at least 2 with n1 endvertices.