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

Published online by Cambridge University Press:  20 November 2018

Svatopluk Poljak  and
Daniel Turzík
Svatopluk Poljak
Affiliation:
Technical University, Prague, Czechoslovakia
Daniel Turzík
Affiliation:
Institute of Chemical Technology, Prague, Czechoslovakia
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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(V3) 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 x1, …, xn.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 519 - 524
Copyright
Copyright © Canadian Mathematical Society 1982

References

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