On the total domination number of graphs

Abstract

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 γ_t(G) is the minimum cardinality of the total dominating sets. In this paper, we will prove that γ_t(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 γ_t(G) ≤ n/2 if the minimum degree of G is at least three.