The shrinking-and-expanding method for the graph enumeration

Several problems on the graph enumeration, which can be solved by the use of a unified method suggested and developed by the first of the authors, are considered. In order to enumerate graphs of a given type, an induced subgraph with particular structure properties should be chosen in each graph and shrunk to a special vertex. The graphs obtained, which contain a fixed (special) vertex of some degree, and also the shrunk subgraphs are enumerated separately by some known methods of graph enumeration. The enumeration of the initial graphs is completed by summing, over all possible degrees of the special vertex, the products of the number of the shrunk subgraphs, the number of the graphs obtained after shrinking, and the number of ways of reconstructing (expanding to) the initial graph.