An introduction to the twin signed total k-domination numbers in directed graphs

Abstract

Let D = (V,A ) be a finite simple directed graph (shortly digraph), N − (v ) and N + (v ) denote the set of in-neighbors and out-neighbors of a vertex v ∈ V , respectively. A function f :V − → { − 1,1 } is called a twin signed total k -dominating function (TSTk DF) if ∑ u ∈ (N − (v ))f (u ) ≥ k and ∑ u ∈ (N + (v ))f (u ) ≥ k for each vertex v ∈ V . The twin signed total k -domination number of D is γ ∗ stk (D ) = min{ω (f ) | f is a TSTk DF of D }, where ω (f ) = ∑ v ∈ V f (v ) is the weight of f . In this paper, we initiate the study of twin signed total k -domination in digraphs and present different bounds on γ ∗ stk (D ). In addition, we determine the twin signed total k -domination number of some classes of digraphs. Our results are mostly extensions of well-known bounds of the twin signed total domination numbers of directed graphs.