### Nordhaus-Gaddum Type Results for Total Domination

*Michael Anthony Henning, Ernst Joubert, Justin Southey*

#### Abstract

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the
sum or product of a parameter of a graph and its complement. In this
paper we study Nordhaus-Gaddum-type results for total domination. We
examine the sum and product of
γ

_{t}(G_{1}) and γ_{t}(G_{2}) where G_{1}⊕G_{2}= K(s,s), and γ_{t}is the total domination number. We show that the maximum value of the sum of the total domination numbers of G_{1}and G_{2}is 2s+4, with equality if and only if G_{1}= sK_{2}or G_{2}= sK_{2}, while the maximum value of the product of the total domination numbers of G_{1}and G_{2}is max{8s,⌊(s+6)^{2}/4 ⌋}.Full Text: PDF PostScript