An Upper Bound for the Total Domination Subdivision Number of a Graph

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number$${{\rm sd}_{\gamma_t}(G)}$$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper, we prove that $${{\rm sd}_{\gamma_t}(G)\leq 2\gamma_t(G)-1}$$ for every simple connected graph G of order n ≥ 3.