Restricted total domination in graphs

The k-restricted total domination number of a graph G is the smallest integer r_k(G, γ_t) such that given any subset U of k vertices of G, there exists a total dominating set of G of cardinality at most r_k(G, γ_t) 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 must be included in the total dominating set. When k = 0, the k-restricted total domination number is the total domination number. In this paper we establish upper bound on the k-restricted total domination number of a connected graph in terms of the order and the size of the graph.