Abstract
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 γ t ( G ) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd γ 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 δ and maximum degree Δ . We prove that if each component of G and G ¯ has order at least 3 and G , G ¯ ≠ C 5 , then γ t ( G ) + γ t ( G ¯ ) ⩽ 2 n 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 γ t ( G ) + sd γ t ( G ¯ ) ⩽ 2 n 3 + 2 . We also give a result on γ t ( G ) + γ t ( G ¯ ) stronger than a conjecture by Harary and Haynes.
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