Twin signed domination numbers in directed graphs

Let \(D=(V,A)\) be a finite simple directed graph (shortly digraph). A function \(f:V\longrightarrow \{-1,1\}\) is called a twin signed dominating function (TSDF) if \(f(N^-[v])\ge 1\) and \(f(N^+[v])\ge 1\) for each vertex \(v\in V\). The twin signed domination number of \(D\) is \(\gamma_{s}^*(D)=\min\{\omega(f)\mid f \text{ is a TSDF of } D\}\). In this paper, we initiate the study of twin signed domination in digraphs and we present sharp lower bounds for \(\gamma_{s}^*(D)\) in terms of the order, size and maximum and minimum indegrees and outdegrees. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs.