Reduced constants for simple cycle graph separation

If G is an n vertex maximal planar graph and delta less than or equal to 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- delta )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 ( square root 2 delta + square root 2-2 delta ) square root n+O(1) vertices. Specifically, when delta identical with 1/3, the separator C is of size ( square root 2/3+ square root 4 /3) square root n+O(1), which is roughly 1.97 square root n. The constant 1.97 is an improvement over the best known so far result of Miller 2 square root 2 approximately 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- delta , 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 square root n+O(1) vertices.