On the differences between the upper irredundance, upper domination and independence numbers of a graph Original Research Article

Let G=(V,E) be a graph and β, Γ and IR its independence, upper domination and upper irredundance number, respectively. We prove that for every l⩾3 there are l-regular graphs for which the difference IR−Γ is arbitrarily large. The case l=3 disproves a conjecture of Henning and Slater (Discrete Math. 158 (1996) 87–98). Furthermore, we present results on the differences IR−β, Γ−β and IR−Γ for general graphs and graphs with restricted maximum degree.