Deciding the Bell number for hereditary graph properties

TL;DR

This paper fully characterizes the family of hereditary graph classes whose growth rate reaches the Bell number and provides an algorithm to decide this property for classes defined by finite forbidden subgraphs.

Contribution

It offers a complete characterization of the minimal classes with speed at least the Bell number and introduces an algorithm to determine this for finite forbidden subgraph sets.

Findings

Complete characterization of minimal classes with speed ≥ Bell number

Algorithm to decide if a hereditary class's speed is above or below the Bell number

Speed is above Bell number for classes with infinitely many minimal forbidden subgraphs

Abstract

The paper [J. Balogh, B. Bollob\'{a}s, D. Weinreich, A jump to the Bell number for hereditary graph properties, J. Combin. Theory Ser. B 95 (2005) 29--48] identifies a jump in the speed of hereditary graph properties to the Bell number $B_n$ and provides a partial characterisation of the family of minimal classes whose speed is at least $B_n$. In the present paper, we give a complete characterisation of this family. Since this family is infinite, the decidability of the problem of determining if the speed of a hereditary property 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 $\mathcal{F}$ of graphs, decides whether the speed of the class of graphs containing no induced subgraphs from the set $\mathcal{F}$ is above or below the Bell number. For properties defined by infinitely many minimal forbidden induced subgraphs, the speed is known to be above the Bell number.