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 γk(G), the connected k-domination number γkc(G); the k-independent domination number γki(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)=γk(G)=γki(G), and that for k⩾2, γkc(G)=1 if irk(G)=1, γkc(G)⩽max{(2k+1)irk(G)-2k,52irk(G)k-72k+2} if irk(G) is odd, and γkc(G)⩽52irk(G)k-3k+2 if irk(G) is even, which generalize some known results.