On the total domination number of graphs - Lam, Peter Che Bor; Wei, Bing (Utilitas Mathematica, 2007)
Let G be a graph of order n with minimum degree at least two and S 2 be a vertex set consisting of all vertices of degree two in G. We call a vertex set T a total dominating set if for any vertex u ∈ V(G) there exists some vertex v ∈ T such that u ∈ N(v). The total domination number denoted by γt(G) is the minimum cardinality of the total dominating sets. In this paper, we will prove that γt(G) ≤ n/2 if the length of the longest paths in the subgraph induced by S2 is at most one. As a consequence, we have that γt(G) ≤ n/2 if the minimum degree of G is at least three.