Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$, denoted by $A_R$, is a graph whose vertex set is the set of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $\mathrm{Ann}(I)\cap \mathrm{Ann}(J)=(0)$. In this paper, we characterize all Artinian rings for which both of the graphs $A_R$ and $\overline{A_R}$ (the complement of $A_R$), are chordal. Moreover, all Artinian rings whose $A_R$ (and thus $\overline{A_R}$) is perfect are characterized.