A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem

Abstract

Let G be a symmetric connected graph without loops. Denote by b ( G ) the maximum number of edges in a bipartite subgraph of G . Determination of b ( G ) is polynomial for planar graphs ([ 6 ], [ 8 ]); in general it is an NP-complete problem ([ 5 ]). Edwards in [ 1 ], [ 2 ] found some estimates of b ( G ) which give, in particular, for a connected graph G of n vertices and m edges, where and ﹛ x ﹜ denotes the smallest integer ≧ x. We give an 0 ( V 3 ) algorithm which for a given graph constructs a bipartite subgraph B with at least f ( m , n ) edges, yielding a short proof of Edwards’ result. Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x 1 , …, x n .