Twin Signed k-Domination Numbers in Directed Graphs

Abstract

‎Let $D=(V,A)$ be a finite simple directed graph (digraph)‎. ‎A‎‎

‎function $f:V\longrightarrow \{-1,1\}$ is called a twin ‎signed‎‎

‎$k$-dominating function (TS$k$DF) if $f(N^-[v])\ge k$ ‎and‎‎

‎$f(N^+[v])\ge k$ for each vertex $v\in V$‎. ‎The twin ‎signed‎‎

‎$k$-domination number of $D$ ‎is‎‎

‎$\gamma_{sk}^*(D)=\min\{\omega(f)\mid f \mbox{ is a ‎TS‎$k$DF of‎ }

‎D\}$‎. ‎In this paper‎, ‎we initiate the study of twin ‎signed‎‎

‎$k$-domination in digraphs and present some bounds ‎on‎‎

‎$\gamma_{sk}^*(D)$ in terms of the order‎, ‎size and ‎maximu‎m and‎

‎minimum indegrees and outdegrees‎, ‎generalising some of ‎the‎‎

‎existing bounds for the twin signed domination numbers in ‎di‎graphs‎

‎and the signed $k$-domination numbers in graphs‎.

‎In addition‎, ‎we determine ‎the‎‎

‎twin signed $k$-domination numbers of some classes of ‎digra‎phs‎.