, Volume 63, Issue 3, pp 645–671

Counting Hexagonal Patches and Independent Sets in Circle Graphs


DOI: 10.1007/s00453-011-9561-y

Cite this article as:
Bonsma, P. & Breuer, F. Algorithmica (2012) 63: 645. doi:10.1007/s00453-011-9561-y


A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the enumeration of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem. It is also shown how to subsequently generate hexagonal patches.


Counting problem Planar graph Circle graph Fullerene Hexagonal patch Fusene Polyhex 

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany

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