### Total global neighbourhood domination

#### Abstract

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*\(2, n - 1, n\).#### Keywords

semi complete graph, total dominating set, connected dominating set

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