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 neigh...
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| Datum: | 2018 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/86 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | 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 \(2, n - 1, n\). |
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