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 t(G) is the minimum cardinality of a total dominating set ofG. The total domination subdivision number sd 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 and maximum degree . We prove that if each component of G and G has order at least 3 and G,G = C5, then t(G) + t(G) 2n 3 + 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 t (G)+ sd t (G) 2n 3 + 2. We also give a result on t(G)+ t(G) stronger than a conjecture by Harary and Haynes. © 2007 Elsevier B.V. All rights reserved.