  A pair of non-adjacent edges is said to be separated in a circular ordering
of vertices, if the endpoints of the two edges do not alternate in the
ordering. The circular separation dimension of a graph $G$, denoted by
$\pi^\circ(G)$, is the minimum number of circular orderings of the vertices of
$G$ such that every pair of non-adjacent edges is separated in at least one of
the circular orderings. This notion is introduced by Loeb and West in their
recent paper. In this article, we consider two subclasses of planar graphs,
namely $2$-outerplanar graphs and series-parallel graphs. A $2$-outerplanar
graph has a planar embedding such that the subgraph obtained by removal of the
vertices of the exterior face is outerplanar. We prove that if $G$ is
$2$-outerplanar then $\pi^\circ(G) = 2$. We also prove that if $G$ is a
series-parallel graph then $\pi^\circ(G) \leq 2$.
