A note on zero-divisor graph of amalgamated duplication of a ring along an ideal

Let R be a commutative ring and I be a non-zero ideal of R. Let R⋈I be the subring of R×R consisting of the elements (r,r+i) for r∈R and i∈I. In this paper we characterize all isomorphism classes of finite commutative rings R with identity and ideal I such that Γ(R⋈I) is planar. We determine the number of vertices of Γ(R⋈I), a necessary and sufficient condition for the graph Γ(R⋈I) to be outerplanar and the domination number of Γ(R⋈I).