Super Restrained Domination in the Join of Some Graphs

Abstract

Let $G=(V(G),E(G))$ be a simple graph. A set $S\subseteq V(G)$ is a restrained dominating set $S$ if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V(G)\mathrm{\setminus }S$. It is a super restrained dominating set if for every vertex $u\in V(G)\mathrm{\setminus }S$, there exists $v\in S$ such that $N_G(v)\cap (V(G)\mathrm{\setminus }S)=\{u\}$. The minimum cardinality of a super restrained dominating set in $G$, denoted by ${\gamma }_{\mathit{spr}}(G)$, is called the super restrained domination number of $G$. In this paper, the researchers obtained the super restrained domination number of the following graphs: $F_n\cong K_1+P_n$, $W_n\cong K_1+C_n$, $S_n\cong K_1+{\bar{K}}_n$, $D_n^{(m)}\cong K_1+m{K}_{n-1}$ and $K_{m,n}\cong {\bar{K}}_m+{\bar{K}}_n$.