We have systematized possible ways of forming the isomers of midi-fullerenes, namely C22, C24, C26 and C28. There are three the most natural mechanisms of their obtaining: embedding carbon dimers into initial fullerenes; fusion of carbon cupolas having the same symmetry; fusion of fullerenes having compatible symmetry. The minimum energy fullerenes are obtained through the use of the first mechanism. It is worth noting that these fullerenes conserve the topological three-fold symmetry. At the same time the graph representation shows the change of symmetry from three-fold symmetry to six-fold one. Combined with the graph analysis, this geometric modeling allows obtain a clear knowledge of the structure of the fullerenes formed.