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

A set S   of vertices of a graph G=(V,E)G=(V,E) with no isolated vertex is a total dominating set   if every vertex of V(G)V(G) is adjacent to some vertex in S. The total domination number  γt(G)γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number  sdγt(G)sdγt(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⩾4n⩾4, minimum degree δδ and maximum degree ΔΔ. We prove that if each component of G   and G¯ has order at least 3 and G,G¯≠C5, then γt(G)+γt(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 sdγt(G)+sdγt(G¯)⩽2n3+2. We also give a result on γt(G)+γt(G¯) stronger than a conjecture by Harary and Haynes.