On the differences of the independence, domination and irredundance parameters of a graph

In this paper we present upper bounds on the differences between the independence, domination and irredundance parameters of a graph. For example, using the Brooks theorem on the chromatic number, we show that for any graph G of order n with maximum degree ∆ ≥ 2 IR(G)− β(G) ≤ ⌊ ∆− 2 2∆ n ⌋ , where β(G) and IR(G) are the independence number and the upper irredundance number of a graph G, respectively. This bound implies a conjecture posed by Rautenbach (Discrete Math. 203 (1999), 239–252).