Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k (G), the connected k-domination number γ_k^c (G); the k-independent domination number γ_kⁱ (G) and the k-irredundance number ir_k (G). The authors prove that if an ir_k-set X is a k-independent set of G, then ir_k (G) = γ_k (G) = γ_kⁱ (G), and that for k ≥ 2, γ_k^c (G) = 1 if ir_k (G) = 1, γ_k^c (G) ≤ max { (2 k + 1) ir_k (G) - 2 k, frac(5, 2) ir_k (G) k - frac(7, 2) k + 2 } if ir_k (G) is odd, and γ_k^c (G) ≤ frac(5, 2) ir_k (G) k - 3 k + 2 if ir_k (G) is even, which generalize some known results.