  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.
  Recently, Abrishami and Henning proved that if $G$ is a cubic graph with
girth at least $6$, then $i(G) \le \frac{4}{11}|V(G)|$. We show a result that
not only improves upon the upper bound of the aforementioned result, but also
applies to a larger class of graphs, and is also tight. Namely, we prove that
if $G$ is a cubic graph without $4$-cycles, then $i(G) \le \frac{5}{14}|V(G)|$,
which is tight. Our result also implies that every cubic graph $G$ without
$4$-cycles satisfies $\frac{i(G)}{\gamma(G)} \le \frac{5}{4}$, which partially
answers a question by O and West in the affirmative.
