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 \(V(G)\backslash S\). It is a super restrained dominating set if for every vertex \(u\in V(G)\backslash S\), there exists \(v\in S\) such that \(N_G(v)\cap (V(G)\backslash S)=\{u \}\). 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: \(F_n\cong K_1+P_n\), \(W_n\cong K_1+C_n\), \(S_n\cong K_1+\overline{K}_n\), \(D^{(m)}_n\cong K_1+mK_{n-1}\) and \(K_{m,n}\cong \overline{K}_m+\overline{K}_n\).