In the context of a commutative ring with unity, denoted as \(\mathcal {S}\) , and its associated set of annihilating ideals \(A(\mathcal {S})\) , there exists a graph known as the extended sum annihilating-ideal graph, denoted as \(AG_\varOmega (\mathcal {S})\) . This graph has its vertex set derived from the set \(A(\mathcal {S})^*\) , and it exhibits a specific pattern of connections between its vertices. More precisely, two distinct vertices, referred to as \(\Im _1\) and \(\Im _2\) , are linked by an edge if and only if one of the following conditions holds: either \(\Im _1\Im _2 = 0\) or \(\Im _1 + \Im _2 \in A(\mathcal {S})\) . In the following research paper, we delve into the classification of Artinian commutative rings, denoted as \(\mathcal {S}\) , with a particular focus on those where the extended sum annihilating-ideal graph takes on one of three distinct forms: a double toroidal graph, a projective plane, or a Klein bottle.