This research article focuses on the co-intersection graph of a commutative ring R. The co-intersection graph of R is a simple graph with vertices that are non-trivial ideals of R, and two distinct vertices I and J are adjacent if and only if I + J, R. The objective of the research article is to characterize the Artinian commutative rings R, determining whether their associated co-intersection graph is tree, bipartite, planar, outerplanar, or toroidal. In essence, the article explores the inherent properties of the co-intersection graph and investigates how specific characteristics of Artinian commutative rings influence the resulting graph structures.