On Distance Connected Domination Numbers of Graphs

Let k be a positive integer and G = (V,E) be a connected graph of order n. A set D ⊆ V is called a k-dominating set of G if each x ∈ V (G) − D is within distance k from some vertex of D. A connected k-dominating set is a k-dominating set that induces a connected subgraph of G. The connected k-domination number of G, denoted by γ k(G), is the minimum cardinality of a connected k-dominating set. Let δ and ∆ denote the minimum and the maximum degree of G, respectively. This paper establishes that γ k(G) ≤ max{1, n − 2k − ∆ + 2}, and γ k(G) ≤ (1+oδ(1))n ln[m(δ+1)+2−t] m(δ+1)+2−t , where m = d k 3 e, t = 3d k 3 e−k, and oδ(1) denotes a function that tends to 0 as δ → ∞. The later generalizes the result of Caro et al’s in [Connected domination and spanning trees with many leaves. SIAM J. Discrete Math. 13 (2000), 202-211] for k = 1.