  Dominating broadcasting is a domination-type structure that models a
transmission antenna network. In this paper, we study a limited version of this
structure, that was proposed as a common framework for both broadcast and
classical domination. In this limited version, the broadcast function is upper
bounded by an integer $k$ and the minimum cost of such function is the
dominating $k$-broadcast number. Our main result is a unified upper bound on
this parameter for any value of $k$ in general graphs, in terms of both $k$ and
the order of the graph. We also study the computational complexity of the
associated decision problem.
