The Domination Parameters of Cubic Graphs

Abstract

Let ir(G), γ(G), i(G), β₀(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any nonnegative integers k₁, k₂, k₃, k₄, k₅ there exists a cubic graph G satisfying the following conditions: γ(G) – ir(G) ≥ k₁, i(G) – γ(G) ≥ k₂, β₀(G) – i(G) > k₃, Γ(G) – β₀(G) – k₄, and IR(G) – Γ(G) – k₅. This result settles a problem posed in [9].