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

The paper [J. Balogh, B. Bollobas, 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 for properties defined by finitely many forbidden induced subgraphs. In other words, we show 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.

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