  Let $R$ be a ring with unity. The upper ideal relation graph $\Gamma_U(R)$ of
the ring $R$ is a simple undirected graph whose vertex set is the set of all
non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and
only if there exists a non-unit element $z \in R$ such that the ideals $(x)$
and $(y)$ contained in the ideal $(z)$. In this article, we classify all the
non-local finite commutative rings whose upper ideal relation graphs are split
graphs, threshold graphs and cographs, respectively. In order to study
topological properties of $\Gamma_U(R)$, we determine all the non-local finite
commutative rings $R$ whose upper ideal relation graph has genus at most $2$.
Further, we precisely characterize all the non-local finite commutative rings
for which the crosscap of $\Gamma_U(R)$ is either $1$ or $2$.
