Typical Structure of Hereditary Graph Families. I. Apex‐free Families

ABSTRACT A family of graphs is hereditary if is closed under isomorphism and taking induced subgraphs. The speed of is the sequence , where denotes the set of graphs in with the vertex set . Alon et al. (2011) gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family there exist and an integer such that The main result of this paper gives a more precise description of typical structure for a restricted class of hereditary families. As a consequence we characterize hereditary families with the speed just above the threshold , generalizing a result of Balogh and Butterfield (2011).