The diagrams, describing the process of forming mini-fullerenes of single carbon atoms and carbon dimers, are suggested. The diagrams have much in common for different fullerenes. Forming fullerenes incorporates several stages. In the first stage an initial reaction-active single atom adds another atom and activates it. In its turn the second atom adds two other atoms creating a chain reaction whereupon mutual turning of the covalent bonds generated leads to folding and to giving rise further covalent bonds that results in fullerene formation. Another way consists in the following. Here a reaction-active carbon dimer adds at once two atoms which create two covalent pairs connected with the dimer. Thereafter the chain reaction is going in two opposite directions similar to the previous case. In both cases the process consists of several stages which differ, but the final configuration, due to folding, is one and the same fullerene. We have constructed graphs for all the mini-fullerenes considered. This approach simplify the analysis both the ways of fullerene forming and its structure.