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Counting Hexagonal Patches and Independent Sets in Circle Graphs

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Abstract

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.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Sekr. MA 5-1, Straße des 17. Juni 136, 10623, Berlin, Germany

    Paul Bonsma

  2. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Felix Breuer

Authors
  1. Paul Bonsma
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  2. Felix Breuer
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Correspondence to Paul Bonsma.

Additional information

An extended abstract of this paper appeared in the proceedings of LATIN 2010.

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Bonsma, P., Breuer, F. Counting Hexagonal Patches and Independent Sets in Circle Graphs. Algorithmica 63, 645–671 (2012). https://doi.org/10.1007/s00453-011-9561-y

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  • DOI: https://doi.org/10.1007/s00453-011-9561-y

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