Finding Fullerene Patches in Polynomial Time
We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is called the boundary code of G. We show that the question whether a given sequence S is a boundary code of some fullerene patch can be answered in polynomial time when such patches have at most five 5-faces. We conjecture that our algorithm gives the correct answer for any number of 5-faces, and sketch how to extend the algorithm to the problem of counting the number of different patches with a given boundary code.
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- 1.Bonsma, P., Breuer, F.: Counting hexagonal patches and independent sets in circle graphs (2009), http://arxiv.org/abs/0808.3881v2
- 2.Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time (2009), http://arxiv.org/abs/0907.2627
- 4.Brinkmann, G., Coppens, B.: An efficient algorithm for the generation of planar polycyclic hydrocarbons with a given boundary. MATCH Commun. Math. Comput. Chem. (2009)Google Scholar
- 6.Brinkmann, G., Fowler, P.W.: A catalogue of growth transformations of fullerene polyhedra. J. Chem. Inf. Comput. Sci. 43, 1837–1843 (2003)Google Scholar
- 8.Deza, M., Fowler, P.W., Grishukhin, V.: Allowed boundary sequences for fused polycyclic patches and related algorithmic problems. J. Chem. Inf. Comput. Sci. 41, 300–308 (2001)Google Scholar
- 11.Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar