  We study the well-known problem of combinatorial classification of
fullerenes. By a (mathematical) fullerene we mean a convex simple three
dimensional polytope with all facets pentagons and hexagons. We analyse
approaches of construction of arbitrary fullerene from the dodecahedron (a
fullerene $C_{20}$). A growth operation is a combinatorial operation that
substitutes the patch with more facets and the same boundary for the patch on
the surface of a simple polytope to produce a new simple polytope. It is known
that an infinite set of different growth operations transforming fullerenes
into fullerenes is needed to construct any fullerene from the dodecahedron. We
prove that if we allow a polytope to contain one exceptional facet, which is a
quadrangle or a heptagon, then a finite set of growth operation is sufficient.
We analyze pairs of objects: a finite set of operations, and a family of
acceptable polytopes containing fullerenes such that any polytope of the family
can be obtained from the dodecahedron by a sequence of operations from the
corresponding set. We describe explicitly three such pairs. First two pairs
contain seven operations, and the last -- eleven operations. Each of these
operations corresponds to a finite set of growth operations and is a
composition of edge- and two edges-truncations.
