Let Λ denote a commutative ring with unity and D (Λ) denote a collection of all annihilating ideals from Λ. An annihilator intersection graph of Λ is represented by the notation AIG (Λ) . This graph is not directed in nature, where the vertex set is represented by D (Λ) * . There is a connection in the form of an edge between two distinct vertices ς and ϱ in AIG (Λ) iff A n n (ς ϱ) ≠ A n n (ς) ∩ A n n (ϱ) . In this work, we begin by categorizing commutative rings Λ , which are finite in structure, so that AIG (Λ) forms a star graph/2-outerplanar graph, and we identify the inner vertex number of AIG (Λ) . In addition, a classification of the finite rings where the genus of AIG (Λ) is 2, meaning AIG (Λ) is a double-toroidal graph, is also investigated. Further, we determine Λ , having a crosscap 1 of AIG (Λ) , indicating that AIG (Λ) is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two. [ABSTRACT FROM AUTHOR]

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