SIGNED STAR k-DOMINATION AND k-DOMATIC NUMBERS OF DIGRAPHS

Abstract

Let D be a simple digraph with vertex set V (D) and arc set A(D), and let k be a positive integer. A function f , A(D) →{-1, 1} is said to be a signed star k-dominating function (SSkD function) on D if ∑a∈A(v) f(a) ≥ k for every vertex v of D, where A(v) is the set of arcs with head v. The signed star k-domination number of a digraph D is defined by γkSS(D) = min{∑a∈2A(D) f(a) | f is a SSkD function on D}. A set {f1, f2, ... , fd} of distinct signed star k-dominating functions on D with the property that ∑d i=1 fi(a) ≤| 1 for each arc a∈A(D), is called a signed star k-dominating family (of functions) on D. The maximum number of functions in a signed star k-dominating family on D is the signed star k-domatic number of D, denoted by dkSS(D). In this paper we study properties of the signed star k-domination and k-domatic number dkSS(D) of digraphs D. Some of our results extend these one given by Atapour, Sheikholeslami, Ghameshlou and Volkmann [1] for the the signed star domatic number, Sheikholeslami and Volkmann [5] for the the signed star k-domatic number and Xu [8] for the signed star domination number of a graph.