Anti-duplication self vertex switching in some graphs

Abstract

For a finite undirected simple graph $G(V, E)$, duplication of a vertex $v \in V(G)$ forms a new graph $G^{\prime}$ by introducing a new vertex $v^{\prime}$ such that $N_{G^{\prime}}\left(v^{\prime}\right)=N_G(v)$. We define anti-duplication of a vertex $v$ in $G$ by introducing a new vertex $v^{\prime}$ which produces a new graph $G^{\prime}$ such that $N_{G^{\prime}}\left(v^{\prime}\right)=\left[N_G(v)\right]^c$. In this paper, we find the number adss $(G)$ when $G$ is $P_n, C_n, K_n, K_{n, m}, S_n, B_n$ and $D(n, m)$.