  A $k$-dominating set is a set $D$ of nodes of a graph such that, for each
node $v$, there exists a node $w \in D$ at distance at most $k$ from $v$. Our
aim is the deterministic distributed construction of small $T$-dominating sets
in time $T$ in networks modeled as undirected $n$-node graphs and under the
$\cal{LOCAL}$ communication model.
  For any positive integer $T$, if $b$ is the size of a pairwise disjoint
collection of balls of radii at least $T$ in a graph, then $b$ is an obvious
lower bound on the size of a $T$-dominating set. Our first result shows that,
even on rings, it is impossible to construct a $T$-dominating set of size $s$
asymptotically $b$ (i.e., such that $s/b \rightarrow 1$) in time $T$.
  In the range of time $T \in \Theta (\log^* n)$, the size of a $T$-dominating
set turns out to be very sensitive to multiplicative constants in running time.
Indeed, it follows from \cite{KP}, that for time $T=\gamma \log^* n$ with large
constant $\gamma$, it is possible to construct a $T$-dominating set whose size
is a small fraction of $n$. By contrast, we show that, for time $T=\alpha
\log^* n $ for small constant $\alpha$, the size of a $T$-dominating set must
be a large fraction of $n$.
  Finally, when $T \in o (\log^* n)$, the above lower bound implies that, for
any constant $x<1$, it is impossible to construct a $T$-dominating set of size
smaller than $xn$, even on rings. On the positive side, we provide an algorithm
that constructs a $T$-dominating set of size $n- \Theta(T)$ on all graphs.
