Total global neighbourhood domination

A subset $D$ of the vertex set of a connected graph $G$ is called a total global neighbourhood dominating set ($\mathrm{tgnd}$-set) of $G$ if and only if $D$ is a total dominating set of $G$ as well as $G^{N}$, where $G^{N}$ is the neighbourhood graph of $G$. The total global neighbourhood domination number ($\mathrm{tgnd}$-number) is the minimum cardinality of a total global neighbourhood dominating set of $G$ and is denoted by $\gamma_{\mathrm{tgn}}(G)$. In this paper sharp bounds for $\gamma_{\mathrm{tgn}}$ are obtained. Exact values of this number for paths and cycles are presented as well. The characterization result for a subset of the vertex set of $G$ to be a total global neighbourhood dominating set for $G$ is given and also characterized the graphs of order $n(\geq 3)$ having $\mathrm{tgnd}$-numbers $2, n - 1, n$.