Total domination in complements of graphs containing no \(K _{4,4}\) (Q1613536)

For a graph \(G= (V,E)\), a set \(S\) is called a total dominating set if, for every \(v\in V\), \(N(v)\cap S\neq\varnothing\). The total domination number is the cardinality of a minimum total dominating set. The authors prove that the total domination number of a graph \(G\) whose complement \(\overline G\) does not contain \(K_{3,3}\) is at most the chromatic number of \(\overline G\) except for complete graphs and graphs belonging to a certain family, which is characterized.