Lower bounds for the total (distance) k-domination number of a graph

For k ≥ 1 and a graph G without isolated vertices, a total (distance) k-dominating set of G is a set of vertices S ⊆ V(G) such that every vertex in G is within distance k to some vertex of S other than itself. The total (distance) k-domination number of G is the minimum cardinality of a total k-dominating set in G, and is denoted by γₖᵗ(G). When $k=1$, the total k-domination number reduces to the total domination number, written γₜ(G); that is, γₜ(G) = γ₁ᵗ(G). This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total (distance) k-domination.