On the ratios between packing and domination parameters of a graph

The relationship ρ_L (G) ≤ ρ (G) ≤ γ (G) between the lower packing number ρ_L (G), the packing number ρ (G) and the domination number γ (G) of a graph G is well known. In this paper we establish best possible bounds on the ratios of the packing numbers of any (connected) graph to its six domination-related parameters (the lower and upper irredundance numbers i r and I R, the lower and upper independence numbers i and β, and the lower and upper domination numbers γ and Γ). In particular, best possible constants a_θ, b_θ, c_θ and d_θ are found for which the inequalities a_θ θ (G) ≤ ρ_L (G) ≤ b_θ θ (G) and c_θ θ (G) ≤ ρ (G) ≤ d_θ θ (G) hold for any connected graph G and all θ ∈ {i r, γ, i, β, Γ, I R}. From our work it follows, for example, that ρ_L (G) ≤ frac(3, 2) i r (G) and ρ (G) ≤ frac(3, 2) i r (G) for any connected graph G, and that these inequalities are best possible.