Deciding the Bell Number for Hereditary Graph Properties - (Extended Abstract)

A graph property is a set of graphs closed under isomorphism. A property is hereditary if it is closed under taking induced subgraphs. Given a graph property X , we write Xn for the number of graphs in X with vertex set {1, 2, . . . , n} and following [1] we call Xn the speed of the property X . The paper [2] identifies a jump in the speed of hereditary graph properties to the Bell number Bn and provides a partial characterization of the family of minimal classes whose speed is at least Bn. In the present work we give a complete characterization of this family. Since this family is infinite, the decidability of the problem of determining if the speed of a hereditary class is above or below the Bell number is questionable. We answer this question positively by showing that there exists an algorithm which, given a finite set F of graphs, decides whether the speed of the class of graphs containing no induced subgraphs from the set F is above or below the Bell number. For properties defined by infinitely many minimal forbidden subgraphs, the speed is known to be above the Bell number. By the structural results obtained, it turns out that the boundary of the Bell number is a partial boundary for well-quasi-ordering by the induced subgraph relation. We show that all the classes below the Bell number are defined by finitely many minimal forbidden induced subgraphs and are all well-quasi-ordered by the induced subgraph relation, while the finitely defined classes which are above the Bell number and have finite distinguishing number are not well-quasi-ordered by the induced subgraph relation. This result gives us some insight how one can approach the question of deciding well-quasi-ordering by the induced subgraph relation in its full generality.