In this paper, we study the signed k-domination and its total version in graphs. By a simple uniform approach we give some new upper and lower bounds on these two parameters of a graph in terms of several different graph parameters. In this way, we can improve and generalize some results in literature. Moreover, we make use of the wellknown theorem of Tur´an [On an extremal problem in graph theory, Math. Fiz. Lapok 48 (1941) 436–452] to bound the signed total k-domination number, γt kS(G), of a Kr+1-free graph G for r ≥ 2.
