Partitioning planar graphs

Reviewer: Diane Mar Spresser

The problem of partitioning the vertices of a planar graph G <__?__Pub Caret1>on n vertices into two sets of prescribed size while cutting a minimum number of edges is considered. For integer s , 0?s?n , an s -partition of G is a partition of the vertex set of G into two sets of size s and n-s . Such a partition is optimal when it cuts the smallest number of edges. The paper is organized into six sections. The first three present an overview of the problem; terminology and definitions; and a table structure, called a slice table, that adapts a known dynamic programming strategy to the partition problem. Sections 4 and 5 contain the paper's main result: an algorithm that finds an optimal s -partition for an n -vertex planar graph G in time O b 2n 32 4.5b , where b is the cutsize (number of edges cut) of an optimal s -partition. The authors derive the corollary (let s= n/2 ) that any planar graph with O log n bisection width may be bisected in polynomial time. Other related results and algorithms are also presented. In Section 6, the authors conclude with a summary of their results and a short list of observations for further study. Overall, this carefully organized paper is a highly readable and clearly written contribution to the existing literature on the graph partitioning problem.