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 \(2, n - 1, n\).