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Let R be a commutative ring with nonzero identity and I be a proper ideal of R. The annihilator graph of R with respect to I, which is denoted by AGI (R), is the undirected graph with vertex-set V (AGI (R)) = {x ∈ R \ I: xy ∈ I for some y ∉ I} and two distinct vertices x and y are adjacent if and only if AI (xy) ≠ = AI (x) ∪ AI (y), where AI (x) = {r ∈ R : rx ∈ I}. In this paper, we study some basic properties of AGI (R), and we characterise when AGI (R) is planar, outerplanar or a ring graph. Also, we study the graph AGI (Zn), where Zn is the ring of integers modulo n. [ABSTRACT FROM AUTHOR]