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

Let G = (V, E) be a graph and beta, Gamma and IR its independence, upper domination and upper irredundance number, respectively. We prove that for every l greater than or equal to 3 there are l-regular graphs for which the difference IR - Gamma 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 - beta, Gamma - beta and IR - Gamma for general graphs and graphs with restricted maximum degree. (C) 1999 Elsevier Science B.V. All rights reserved.