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 k(G), the connected k-domination number c k (G); the k-independent domination number i k (G) and the k-irredundance number irk(G). The authors prove that if an irk-set X is a k-independent set of G, then irk(G) = k(G) = k(G), and that for k 2, c k (G) = 1 if irk(G) = 1, k(G) max{(2k + 1)irk(G) − 2k, 5 2 irk(G)k − 7 2k + 2} if irk(G) is odd, and c k (G) 5 2 irk(G)k − 3k + 2 if irk(G) is even, which generalize some known results. © 2006 Elsevier B.V. All rights reserved.