Finding Fullerene Patches in Polynomial Time

Abstract

Abstract. 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.

Counting Hexagonal Patches and Independent Sets in Circle Graphs

“…After we presented an early version of this work [2], Jack Graver pointed us to a similar well-studied problem in topology. Let S 1 denote the unit circle and D 2 the unit disk in R 2 .…”

“…Until now, the only algorithms known for the FULLERENE PATCH problem were (super)exponential time branching algorithms [7] and/or algorithms for special cases of the problem [8,9,12]. In [3] we gave a polynomial Turing reduction from the problem on instances S with d 2 (S) > d 3 (S) to instances S with d 2 (S ) − d 3 (S ) = 6. When d 2 (S ) − d 3 (S ) = 6, any fullerene patch with boundary code S is a hexagonal patch (Proposition 1).…”

“…, N − 1} can be generated uniformly at random. Combined with the result in [3], our results give a polynomial time algorithm to decide for any given constant N whether at least N fullerene patches exist with a given boundary code S, when d 2 (S) > d 3 (S), and to generate these patches (see [3] for details). Combined with [8] this allows for much more efficient enumeration of fullerenes.…”

Counting Hexagonal Patches and Independent Sets in Circle Graphs

“…The algorithm can however be extended to generate all patches in time k O (1) · O(n), where n is the number of returned solutions. We remark that it is not hard to generalize our result to the generalizations introduced in [8]: A 2-connected plane graph is an (m, k)-patch if all inner faces have length k, inner vertices have degree m and boundary vertices have degree at most m. Our methods work for instance for (4,4) and (6,3)-patches in addition to (3, 6)-patches, but for simplicity we restrict to hexagonal patches.…”

Extending patches to fullerenes

“…It is well-known (see for instance [1]) that the number of pentagons in a patch is equal to 6 − d 2 + d 3 where d i is the number of degree i vertices on the boundary. By definition, the number of degree 2 vertices and the number of degree three vertices on the boundary are the same except on a break edge or a bend edge.…”