  We revisit $k$-Dominating Set, one of the first problems for which a tight
$n^k-o(1)$ conditional lower bound (for $k\ge 3$), based on SETH, was shown
(P\u{a}tra\c{s}cu and Williams, SODA 2007). However, the underlying reduction
creates dense graphs, raising the question: how much does the sparsity of the
graph affect its fine-grained complexity?
  We first settle the fine-grained complexity of $k$-Dominating Set in terms of
both the number of nodes $n$ and number of edges $m$. Specifically, we show an
$mn^{k-2-o(1)}$ lower bound based on SETH, for any dependence of $m$ on $n$.
This is complemented by an $mn^{k-2+o(1)}$-time algorithm for all $k\ge 3$. For
the $k=2$ case, we give a randomized algorithm that employs a Bloom-filter
inspired hashing to improve the state of the art of $n^{\omega+o(1)}$ to
$m^{\omega/2+o(1)}$. If $\omega=2$, this yields a conditionally tight bound for
all $k\ge 2$.
  To study if $k$-Dominating Set is special in its sensitivity to sparsity, we
consider a class of very related problems. The $k$-Dominating Set problem
belongs to a type of first-order definable graph properties that we call
monochromatic basic problems. These problems are the natural monochromatic
variants of the basic problems that were proven complete for the class FOP of
first-order definable properties (Gao, Impagliazzo, Kolokolova, and Williams,
TALG 2019). We show that among these problems, $k$-Dominating Set is the only
one whose fine-grained complexity decreases in sparse graphs. Only for the
special case of reflexive properties, is there an additional basic problem that
can be solved faster than $n^{k\pm o(1)}$ on sparse graphs.
  For the natural variant of distance-$r$ $k$-dominating set, we obtain a
hardness of $n^{k-o(1)}$ under SETH for every $r\ge 2$ already on sparse
graphs, which is tight for sufficiently large $k$.
