Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number@c"t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd"@c"""t(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n>=4, minimum degree @d and maximum degree @D. We prove that if each component of G and G@? has order at least 3 and G,G@? C"5, then @c"t(G)+@c"t(G@?)=<2n3+2 and if each component of G and G@? has order at least 2 and at least one component of G and G@? has order at least 3, then sd"@c"""t(G)+sd"@c"""t(G@?)=<2n3+2. We also give a result on @c"t(G)+@c"t(G@?) stronger than a conjecture by Harary and Haynes.