On the total domination subdivision numbers in graphs

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 sdγt (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. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727-733) proved that for any connected graph G of order n ≥ 3, sdγt (G) ≤ 2γt (G)− 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sdγt (G) ≤ 2α ′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α ′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sdγt (G) = 2γt (G)−1. MSC: 05C69