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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 $γ^+_{maj}(D).$ 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.