Planar Separator Theorem - Lower Bounds

Lower Bounds

In a √n × √n grid graph, a set S of s < √n points can enclose a subset of at most s(s − 1)/2 grid points, where the maximum is achieved by arranging S in a diagonal line near a corner of the grid. Therefore, in order to form a separator that separates at least n/3 of the points from the remaining grid, s needs to be at least √(2n/3), approximately 0.82√n.

There exist n-vertex planar graphs (for arbitrarily large values of n) such that, for every separator S that partitions the remaining graph into subgraphs of at most 2n/3 vertices, S has at least √(4π√3)√n vertices, approximately 1.56√n. The construction involves approximating a sphere by a convex polyhedron, replacing each of the faces of the polyhedron by a triangular mesh, and applying isoperimetric theorems for the surface of the sphere.