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

¹ Department of Mathematics, Faculty of basic sciences, University of Bonab, Bonab, I.R. Iran.
m.atapour@bonabu.ac.ir
² Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, I.R. Iran.
n.dehgardi@sirjantech.ac.ir
³ Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany.
volkm@math2.rwth-aachen.de

Received: 12 November 2016
Accepted: 6 March 2017

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 (TSTkDF) 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 TSTkDF 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.