  Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that
each vertex not in $S$ has a neighbor in $S$. The domination number of $G$,
denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The
independent domination number of $G$, denoted $i(G)$, is the minimum size of a
dominating set of $G$ that is also independent. Note that every graph has an
independent dominating set, as a maximal independent set is equivalent to an
independent dominating set.
  Let $G$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq
4$. Generalizing a result by Lam, Shiu, and Sun, we prove that $i(G)\le
\frac{k-1}{2k-1}|V(G)|$, which is tight for $k = 4$. This answers a question by
Goddard et al. in the affirmative. We also show that $\frac{i(G)}{\gamma(G)}
\le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor,
\v{S}krekovski, and Tepeh. In addition, we prove that a graph $G'$ with maximum
degree at most $4$ satisfies $i(G') \le \frac{5}{9}|V(G')|$, which is also
tight.
