Deciding the Bell number for hereditary graph properties

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.