@Article{Haynes2022,
author = "Haynes, Teresa W. and Henning, Michael A.",
title = "Nordhaus-Gaddum bounds for upper total domination",
journal = "Opuscula Math.",
year = "2022",
volume = "42",
number = "4",
pages = "573--582",
doi = "https://doi.org/10.7494/OpMath.2022.42.4.573",
url = "https://doi.org/10.7494/OpMath.2022.42.4.573",
abstract = "A set \(S\) of vertices in an isolate-free graph \(G\) is a total dominating set if every vertex in \(G\) is adjacent to a vertex in \(S\). A total dominating set of \(G\) is minimal if it contains no total dominating set of \(G\) as a proper subset. The upper total domination number \(\Gamma_t(G)\) of \(G\) is the maximum cardinality of a minimal total dominating set in \(G\). We establish Nordhaus-Gaddum bounds involving the upper total domination numbers of a graph \(G\) and its complement \(\overline{G}\). We prove that if \(G\) is a graph of order \(n\) such that both \(G\) and \(\overline{G}\) are isolate-free, then \(\Gamma_t(G) + \Gamma_t(\overline{G}) \leq n + 2\) and \(\Gamma_t(G)\Gamma_t(\overline{G}) \leq \frac{1}{4}(n+2)^2\), and these bounds are tight.",
keywords = "upper total domination, Nordhaus-Gaddum bounds",
issn = "1232-9274"
}