A study on distance k-domination in digraphs

Let D = ( V, A ) be a finite and simple directed graph (digraph) with vertex set V and arc set A . For an integer k ≥ 1, the out k -neighborhood of a vertex ν is defined as N k + ( ν ) = { u ∈ V ( D ) / d → ( ν , u ) ≤ k } . A set S ⊆ V is said to be a distance k-dominating set in D if N + k ( S ) ∪ S = V . The minimum distance k -dominating set in D is the distance k -domination number, γ k → ( D ) . For two integers s ≥ 2 and k , the (s, k ) - kernal set and the (s, k ) - kernal number, ξ k → ( D ) are studied. This parameter ξ k → ( D ) is also computed for different types of digraphs. Strong digraphs (SDs) are studied and γ k → ( D ) of SDs are determined in terms of the radius and diameter of SDs. Also (2, k ) - kernals of directed wounded spider are classified with ξ k → ( D ) . Some upper and lower bounds on γ k → ( D ) are determined and characterized some digraphs achieving these bounds.