Vertices of degree 6 in a contraction critically 6-connected graph Original Research Article

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 γkc(G), is the minimum cardinality of a connected k-dominating set of G. In this paper, we show that for k⩾2, γkc(G)⩽(2k+1)dkc(Ḡ) if both G and Ḡ are connected, where dkc(G) denotes the connected k-domatic number of G, the maximum number of classes in a partition of V into connected k-dominating sets.