Signed Cycle Domination Numbers of Digraphs

Abstract

The concept of signed cycle domination number of graphs, introduced by B. Xu [B. Xu, On signed cycle domination in graphs, Discrete Math. $309(2009)1007-1012]$, is extended to digraphs, denoted by ${\gamma }_{sc}^{\prime }(D)$ for a digraph $D$. We establish bounds on ${\gamma }_s(D)$, characterize all digraphs $D$ with ${\gamma }_{sc}^{\prime }(D)=|A(D)|-2$, and determine the exact value of ${\gamma }_{sc}^{\prime }(D)$ for specific classes of digraphs $D$. Furthermore, we define the parameter $g^{\prime }(m,n)=min\{{\gamma }_{sc}^{\prime }(D)\mid D\text{ is a digraph with }|V(D)|=n\text{ and }|A(D)|=m\}$ and obtain its value for all integers $n$ and $m$ satisfying $0\le m\le n(n-1)$.