Let
$\gamma(G)$
and
$${\gamma _ \circ }(G)$$
denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then
$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$
In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then
$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$
for some constant
c independent of
d. This result is sharp in the sense that as
$d \rightarrow \infty$
, almost all
d-regular
n-vertex graphs G of girth at least five have
$$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$
Furthermore, if G is a disjoint union of
${n}/{(2d)}$
complete bipartite graphs
$K_{d,d}$
, then
${\gamma_\circ}(G) = \frac{n}{2}$
. We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that
${\gamma_\circ}(G) \sim {n}/{2}$
as
$d \rightarrow \infty$
. Therefore both the girth and regularity conditions are required for the main result.