This research article examines the co-intersection graph of a commutative ring S. The co-intersection graph Ω(S) is defined as a simple graph where the vertices correspond to the non-trivial ideals of S, and two distinct vertices I and J are adjacent if and only if I+J≠S. In this study, we first classify the Artinian rings S for which Ω(S) is isomorphic to certain basic graphs, such as a unicycle, a split graph, or a threshold graph. Subsequently, we investigate the genus of Ω(S) and identify the rings S for which Ω(S) represents a double-toroidal graph. Furthermore, we analyze the crosscap of Ω(S) and determine the rings S for which Ω(S) corresponds to the projective plane or the Klein bottle. Lastly, we compute the book thickness of Ω(S) for cases where the genus is at most one.
Alsuraiheed et al. (Wed,) studied this question.