Abstract
A set D of vertices of a connected graph G=( V, E) is called a connected k-dominating set of G if every vertex in V– D is within distance k from some vertex of D, and the induced subgraph G[ D] is connected, where k⩾1 is an integer. The connected k-domination number of G, denoted by γ k c( G), is the minimum cardinality of a connected k-dominating set of G. In this paper, we show that for k⩾2, γ k c (G)⩽(2k+1)d k c ( G ̄ ) if both G and G ̄ are connected, where d k c( G) denotes the connected k-domatic number of G, the maximum number of classes in a partition of V into connected k-dominating sets.
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