A graded version of the separator theorem for planar graphs

Source abstract

We prove that every planar graph on $n$ vertices has a vertex ordering $v_1,v_2,\ldots,v_n$ in which each prefix $\{v_1,\ldots,v_i\}$ is joined to at most $C\sqrt{i}$ elements of $\{v_{i+1},\ldots,v_n\}$, where $C$ is an absolute constant. Consequently, every prefix can be separated from its complement by deleting $O\left(\sqrt{i}\right)$ vertices lying entirely in its complement.