  A subset $D$ of the vertex set $V$ of a graph $G$ is called an
$[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one
vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the
minimum number of vertices is called a $\gamma_{[1,k]}$-set and the number of
its vertices is the $[1,k]$-domination number $\gamma_{[1,k]}(G)$ of $G$. In
this short note we show that the decision problem whether $\gamma_{[1,k]}(G)=n$
is an $NP$-hard problem, even for bipartite graphs. Also, a simple construction
of a bipartite graph $G$ of order $n$ satisfying $\gamma_{[1,k]}(G)=n$ is given
for every integer $n\geq (k+1)(2k+3)$.
