A Note on Planar Graphs and Circle Orders

A partially ordered set P is called a circle order if one can assign to each element $a \in P$ a circular disk in the plane $C_a $ so that $a < b$ and only if $C_a \subset C_b $. To a graph $G = ( V,E )$ associate a poset $P( G )$ whose elements are the vertices and edges of G. $v < e$ in $P( G )$ exactly when $v \in V, e \in E$, and v is an endpoint of e. It is shown that G is planar if and only if $P( G )$ is a circle order.