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.