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

A set S of vertices of a graph image with no isolated vertex is a total dominating set if every vertex of image is adjacent to some vertex in S. The total domination number image is the minimum cardinality of a total dominating set of G. The total domination subdivision number image is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order image, minimum degree image and maximum degree image. We prove that if each component of G and image has order at least 3 and image, then image and if each component of G and image has order at least 2 and at least one component of G and image has order at least 3, then image. We also give a result on image stronger than a conjecture by Harary and Haynes.