Reduced constants for simple cycle graph separation

Abstract.

 If G is an n vertex maximal planar graph and δ≤1 3, then the vertex set of G can be partitioned into three sets A, B, C such that neither A nor B contains more than (1−δ)n vertices, no edge from G connects a vertex in A to a vertex in B, and C is a cycle in G containing no more than (√2δ+√2−2δ)√n+O(1) vertices. Specifically, when δ=1 3, the separator C is of size (√2/3+√4/3)√n+O(1), which is roughly 1.97√n. The constant 1.97 is an improvement over the best known so far result of Miller 2√2≈2.82. If non-negative weights adding to at most 1 are associated with the vertices of G, then the vertex set of G can be partitioned into three sets A, B, C such that neither A nor B has weight exceeding 1−δ, no edge from G connects a vertex in A to a vertex in B, and C is a simple cycle with no more than 2√n+O(1) vertices.