Let G be a graph of order n with minimum degree at least two and S2 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 w e T such that u e N(v). The total domination number denoted by γyt(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((3) ≤ n/2 if the minimum degree of G is at least three.
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