Restricted total domination in graphs with minimum degree two

The k-restricted total domination number of a graph G is the smallest integer t_k such that given any subset U of k vertices of G, there exists a total dominating set of G of cardinality at most t_k containing U. Hence, the k-restricted total domination number of a graph G measures how many vertices are necessary to totally dominate a graph if an arbitrary set of k vertices are specified to be in the set. When k = 0, the k-restricted total domination number is the total domination number. For 1 ≤ k ≤ n, we show that t_k ≤ 4(n + k)/7 for all connected graphs of order n and minimum degree at least two and we characterize the graphs achieving equality. These results extend earlier results of the author (J. Graph Theory 35 (2000), 21-45). Using these results we show that if G is a connected graph of order n with the sum of the degrees of any two adjacent vertices at least four, then γ_t(G) ≤ 4n/7 unless G ∈ {C ₃, C₅, C₆, C₁₀}.