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. 1

In this paper we consider fullerene patches with nice boundaries containing between one and five pentagonal faces. We find necessary conditions for the side lengths of such patches, and then prove these conditions are sufficient by constructing such patches.

In this paper, we show that fullerene patches with nice boundaries containing between 1 and 5 pentagons fall into several equivalence classes; furthermore, any two fullerene patches in the same class can be transformed into the same minimal configuration using combinatorial alterations.