Majority out-dominating functions in digraphs

At least two different notions have been published under the name "majority domination in graphs": Majority dominating functions and majority dominating sets. In this work we extend the former concept to digraphs. Given a digraph \(D=(V,A),\) a function \(f : V \rightarrow \{-1,1\}\) such that \(f(N^+[v])\geq1\) for at least half of the vertices \(v\) in \(V\) is a majority out-dominating function (MODF) of \(D.\) The weight of a MODF \(f\) is \(w(f)=\sum\limits_{v\in V}f(v),\) and the minimum weight of a MODF in \(D\) is the majority out-domination number of \(D,\) denoted \(\gamma^+_{maj}(D).\) In this work we introduce these concepts and prove some results regarding them, among which the fact that the decision problem of finding a majority out-dominating function of a given weight is NP-complete.