Two forbidden induced subgraphs and well-quasi-ordering

Korpelainen, Nicholas and Lozin, Vadim (2011). Two forbidden induced subgraphs and well-quasi-ordering. Discrete Mathematics, 311(16) pp. 1813–1822.

DOI: https://doi.org/10.1016/j.disc.2011.04.023

Abstract

It is known that a class of graphs defined by a single forbidden induced subgraph G is well-quasi-ordered by the induced subgraph relation if and only if G is an induced subgraph of P₄. However, very little is known about well-quasi-ordered classes of graphs defined by more than one forbidden induced subgraph. We conjecture that for any natural number k, there are finitely many minimal classes of graphs defined by k forbidden induced subgraphs which are not well-quasi-ordered by the induced subgraph relation and prove the conjecture for k=2. We explicitly reveal many of the minimal classes defined by two forbidden induced subgraphs which are not well-quasi-ordered and many of those which are well-quasi-ordered by the induced subgraph relation.

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