Distance irredundance and connected domination numbers of a graph

Abstract

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 k i ( 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 k i ( 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 , 5 2 ir k ( G ) k - 7 2 k + 2 } if ir k ( G ) is odd, and γ k c ( G ) ⩽ 5 2 ir k ( G ) k - 3 k + 2 if ir k ( G ) is even, which generalize some known results.