Transformations of cubic graphs

For simple r-regular graph, an edge-reduction and three transformations (S̃-, X̃-, and H̃-transformations) are defined which preserve the regularity. In the case r = 3, relations between them are discussed and it is proved that for any two connected cubic graphs with the same order one is obtained from the other by a finite sequence of S̃-transformations. Then it defines a metric on the set of connected cubic graphs.