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Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ _I (R), is the graph whose vertices are the set {x ∊ R\I | xy ∊ I for some y ∊ R\I} with distinct vertices x and y adjacent if and only if xy ∊ I. In the case I = 0, Γ₀(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ _I (R) ≅ Γ _J (S) and Γ(R/I) ≅ Γ(S/J). We also discuss when Γ _I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ _I (R).