# Reduced constants for simple cycle graph separation

DOI: 10.1007/s002360050082

- Cite this article as:
- Djidjev, H. & Venkatesan, S. Acta Informatica (1997) 34: 231. doi:10.1007/s002360050082

## 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.