Counting Hexagonal Patches and Independent Sets in Circle Graphs
- 147 Downloads
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.
KeywordsCounting problem Planar graph Circle graph Fullerene Hexagonal patch Fusene Polyhex
Unable to display preview. Download preview PDF.
- 1.Blank, S.J.: Extending immersions of the circle. Ph.D. thesis, Brandeis University, Waltham, MA, USA (1967) Google Scholar
- 2.Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time I: counting hexagonal patches. arXiv:0808.3881v1 (2008)
- 3.Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time. In: Proceedings of the 20th International Symposium on Algorithm and Computation (ISAAC 2009). Lecture Notes in Computer Science, vol. 5878, pp. 750–759. Springer, Berlin (2009) Google Scholar
- 10.Brinkmann, G., Delgado-Friedrichs, O., von Nathusius, U.: Numbers of faces and boundary encodings of patches. In: Graphs and Discovery, Proceedings of the DIMACS Workshop on Computer Generated Conjectures from Graph Theoretic and Chemical Databases. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 69, pp. 27–38. AMS, Providence (2005) Google Scholar
- 16.Eppstein, D., Mumford, E.: Self-overlapping curves revisited. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 160–169. SIAM, Philadelphia (2009) Google Scholar
- 28.Valiente, G.: A new simple algorithm for the maximum-weight independent set problem on circle graphs. In: Proceedings of the 14th International Symposium on Algorithm and Computation (ISAAC 2003). Lecture Notes in Computer Science, vol. 2906, pp. 129–137. Springer, Berlin (2003) Google Scholar