A Note on the Signed Clique Domination Numbers of Graphs

Abstract

Let $G=\left(V,E\right)$ be a graph. A function $f:E⟶\left\{-1,+1\right\}$ is said to be a signed clique dominating function (SCDF) of $G$ if ${\displaystyle {\sum }_{e\in E\left(K\right)}f\left(e\right)\ge 1}$ holds for every nontrivial clique $K$ in $G$ . The signed clique domination number of $G$ is defined as ${\gamma }_{\text{scl}}^{\prime }\left(G\right)=\min \left\{{\displaystyle {\sum }_{e\in E\left(G\right)}f\left(e\right)|f\text{is\hspace{0.17em}an\hspace{0.17em}SCDF\hspace{0.17em}of}G}\right\}$ . In this paper, we investigate the signed clique domination numbers of join of graphs. We correct two wrong results reported by Ao et al. (2014) and Ao et al. (2015) and determine the exact values of the signed clique domination numbers of $P_m\vee \overline{K_n}$ and $C_m\vee K_n$ .