  We describe a new construction algorithm for the recursive generation of all
non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm
stays entirely within the class of IPR fullerenes, that is: every IPR fullerene
is constructed by expanding a smaller IPR fullerene unless it belongs to
limited class of irreducible IPR fullerenes that can easily be made separately.
The class of irreducible IPR fullerenes consists of 36 fullerenes with up to
112 vertices and 4 infinite families of nanotube fullerenes. Our implementation
of this algorithm is faster than other generators for IPR fullerenes and we
used it to compute all IPR fullerenes up to 400 vertices.
