Bounds on the connected [formula omitted]-domination number in graphs

Abstract

Let G = ( V , E ) be a simple graph, and let k be a positive integer. A subset D ⊆ V is a k - dominating set of the graph G if every vertex v ∈ V − D is adjacent to at least k vertices of D . The k - domination number γ k ( G ) is the minimum cardinality among the k -dominating sets of G . A subset D ⊆ V is said to be a connected k - dominating set if D is k -dominating and its induced subgraph is connected. D is called total k - dominating if every vertex in V has at least k neighbors in D and it is a connected total k - dominating set if, additionally, its induced subgraph is connected. The minimum cardinalities of a connected k -dominating set, a total k -dominating set, and a connected total k -dominating set are respectively denoted as γ k c ( G ) , γ k t ( G ) and γ k c , t ( G ) . In this paper, we establish different sharp bounds on the connected k -domination number γ k c ( G ) , involving also the parameters γ k ( G ) , γ k t ( G ) and γ k c , t ( G ) .