an early result in probabilistic combinatorics, due independently to arnautov, payan and lovasz is that if $g$ is a graph on $n$ vertices with minimum degree $\delta$ then $g$ contains a dominating set of size of at most $n(1+ \log(\delta+1))/(\delta+1)$. here, under the further assumptions that $g$ is of girth at least 5 and is $d$-regular we show that, effectively, such a dominating set can be taken to be independent. that is, we show that every $n$-vertex $d$-regular graph of girth 5 contains an independent dominating set of size $(n \log d)/d + o(n/d)$ as $d \to \infty$. we further give examples showing that the girth requirement is necessary and that regularity is necessary in the sense that if $g$ has minimum degree $\delta$, the smallest independent set may be much larger than $(n \log \delta)/\delta$. this is joint work with a. harutyunyan, and j. verstraete.