It is known that the set of all simple graphs is not well-quasi-ordered by the induced subgraph relation, i.e. it contains infinite antichains (sets of incomparable elements) with respect to this relation. However, some particular graph classes are well-quasi-ordered by induced subgraphs. Moreover, some of them are well-quasi-ordered by a stronger relation called labelled induced subgraphs. In this paper, we conjecture that a hereditary class X which is well-quasi-ordered by the induced subgraph relation is also well-quasi-ordered by the labelled induced subgraph relation if and only if X is defined by finitely many minimal forbidden induced subgraphs. We verify this conjecture for a variety of hereditary classes that are known to be well-quasi-ordered by induced subgraphs and prove a number of new results supporting the conjecture.

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The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596-603] and can be defined as follows. A representation of an n-vertex graph G is said to be implicit if it assigns to each vertex of G a binary code of length O(log n) so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an n-vertex graph uses O(n log n) bits, any class of graphs admitting such a representation contains 2 O(n log n) labelled graphs with n vertices. In the terminology of [J. Balogh, B. Bollobás, D. Weinreich, The speed of hereditary properties of graphs, J. Combin. Theory B 79 (2000) 131-156] such classes have at most factorial speed of growth. In this terminology, the implicit graph conjecture can be stated as follows: every class with at most factorial speed of growth which is hereditary admits an implicit representation. The question of deciding whether a given hereditary class has at most factorial speed of growth is far from being trivial. In the present paper, we introduce a number of tools simplifying this question. Some of them can be used to obtain a stronger conclusion on the existence of an implicit representation. We apply our tools to reveal new hereditary classes with the factorial speed of growth. For many of them we show the existence of an implicit representation.