Let R be a commutative ring with identity and 𝒰R be the set of all nonzero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝒰R and two vertices x and y are adjacent whenever ann(x) ∩ ann(y) = (0). In this paper, we characterize all commutative Artinian non-local rings R for which the 𝒞𝒜R has genus one and two. Also we characterize all commutative Artinian non-local rings R for which 𝒞𝒜R has crosscap one. Finally, we characterize all finite commutative non-local rings for which g(Г2(R)) = g(𝒞𝒜R) = 0 or 1.
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