Graphs with large total domination number

Let G= (V,E) be a graph. A set S⊆ Vis a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination number of G, denoted by _γt(G), is the minimum cardinality of a total dominating set of G. We establish a property of minimum total dominating sets in graphs. If G is a connected graph of order n≥3, then (see [3]) _γt,(G)≤2n/3. We show that if G is a connected graph of order n with minimum degree at least 2, then either _γt(G) ≤ 4n/7 or Gε {C₃, C₅, C₆, C₁₀}. A characterization of those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G) ≥ 2 and _γt(G) ≥ 4n/7 is obtained. We establish that if G is a connected graph of size q with minimum degree at least 2, then _γt(G) ≤ (q+2)/2. Connected graphs G of size q with minimum degree at least 2 satisfying _γt(G) ≥ q/2 are characterized.