Crosscap of the ideal based zero-divisor graph

Let RR be a commutative ring and II be an ideal of RR. The ideal based zero-divisor graph, denoted by ΓI(R)ΓI(R), is the graph with the vertex set {x∈R−I:xy∈Ifor somey∈R−I} and two distinct vertices xx and yy are adjacent if and only if xy∈Ixy∈I. In this paper, we classify all finite quotient rings R/IR/I and ideals II of RR for which the crosscap of ΓI(R)ΓI(R) is at most one. Moreover, we investigate certain properties on the crosscap of ΓI(R)ΓI(R) in the general case also.