Constructive Graph Theory: Generation Methods, Structure and Dynamic Characterization of Closed Classes of Graphs -- A survey

TL;DR

This survey explores methods for generating and characterizing closed classes of graphs using binary operations, introducing bases and restrictions that preserve graph properties, with potential applications in graph theory.

Contribution

It provides a comprehensive overview of constructive descriptions of closed graph classes, including bases, restrictions, and their unique properties, advancing the understanding of graph generation methods.

Findings

Each closed class has a unique elemental basis.

Every closed class has at least one operational basis.

Constructive descriptions for classical graph classes are obtained.

Abstract

The processes of constructing some graphs from others using binary operations of union with intersection (gluing) are studied. For graph classes closed with respect to gluing operations the elemental and operational bases are introduced. The generating bases together with the system of restrictions on the gluing operation that preserves the characteristic properties of graphs form constructive descriptions of the closed classes of graphs. It is shown that each closed class of graphs has a unique elemental basis and at least one operational basis. A constructive description of a closed class of all graphs and all bases precompletion closed sub-classes of it are considered. For each of them its characteristic properties and a diagram of the inclusion of sub-classes in super classes is given. Constructive descriptions are obtained for some classes of graphs with classical properties. In conclusion, some possible applications of constructive graph theory are discussed.