  Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$,
denoted by $\Gamma(R)$, is a graph whose vertices are the proper ideals of $R$
which are not contained in the Jacobson radical of $R$, and two vertices $I_1$
and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all
commutative rings whose co-maximal ideal graphs are planar. In 2012 the
following question was posed: If $\Gamma(R)$ is an infinite star graph, can $R$
be isomorphic to the direct product of a field and a local ring? In this paper,
we give an affirmative answer to this question.
