Signed double Roman k-domination in digraphs

Utilitas Mathematica, vol. 128, pp. 109–125

Abstract

Let \(D\) be a finite simple digraph with vertex set \(V(D)\). For \(v\in V(D)\), the set \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). Let \(k\ge 1\) be an integer. A signed double Roman \(k\)-dominating function (SDR\(k\)DF) on a digraph \(D\) is a function \(f:V(D)\rightarrow\{-1,1,2,3\}\) satisfying the following conditions: (i) \(\sum\limits_{x\in N^-[v]}f(x)\ge k\) for each \(v\in V(D)\); (ii) every vertex \(u\) with \(f(u)=-1\) has an in-neighbor \(z\) with \(f(z)=3\) or two in-neighbors \(x\) and \(y\) with \(f(x)=f(y)=2\); (iii) every vertex \(u\) with \(f(u)=1\) has an in-neighbor \(z\) with \(f(z)\ge 2\). The weight of an SDR\(k\)DF \(f\) is \(\omega(f)=\sum\limits_{v\in V(D)}f(v)\). The signed double Roman \(k\)-domination number \(\gamma_{sdR}^k(D)\) is the minimum weight of an SDR\(k\)DF on \(D\). In this paper, we study the signed double Roman \(k\)-domination number of digraphs and present various bounds on \(\gamma_{sdR}^k(D)\). In addition, we determine this parameter for several classes of digraphs. Some of our results extend well-known properties of the signed double Roman \(k\)-domination number \(\gamma_{sdR} ^k(G)\) of graphs \(G\).

Authors 2

1. Lutz Volkmann Aachen Institute for Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Practical Mathematics, RWTH Aachen University, 52056 Aachen, Germany

2. Vadim Zverovich

   University of the West of England

   Affiliation as printed

   Mathematics and Statistics Research Group, University of the West of England, Bristol BS16 1QY, UK