The diagrams, describing the process of forming mini-fullerenes (from C4 to C20) of single carbon atoms and carbon dimers, were suggested by one of the authors elsewhere. In this contribution we solved an inverse problem, i.e. how to predict possible ways of forming mini-fullerenes, if one knows its graphs. We have analyzed the graphs describing the process of forming mini-fullerenes and found that they can be formed not only of single carbon atoms and carbon dimers but also of small carbon clusters. On the basis of the graphs it is possible to distinguish different families of mini-fullerenes and therefore one can make a classification of these unusual carbon structures. In the course of the analysis some innovation to the graph theory was done. We suggested considering a cluster of three or four atoms as a big point (vertex) contrary to a zero-size point (vertex) of a common graph. In this case one obtains a graph, which is identical to a simpler graph. It allows do some operations with this graph in same manner as with a usual graph that simplifies an analysis.