Boundary Properties of Well-Quasi-Ordered Sets of Graphs

Abstract

Let \({\cal Y}_k\) be the family of hereditary classes of graphs defined by k forbidden induced subgraphs. In Korpelainen and Lozin (Discrete Math 311:1813–1822, 2011), it was shown that \({\cal Y}_2\) contains only finitely many minimal classes that are not well-quasi-ordered (wqo) by the induced subgraph relation. This implies, in particular, that the problem of deciding whether a class from \({\cal Y}_2\) is wqo or not admits an efficient solution. Unfortunately, this idea does not work for k ≥ 3, as we show in the present paper. To overcome this difficulty, we introduce the notion of boundary properties of well-quasi-ordered sets of graphs. The importance of this notion is due to the fact that for each k, a class from \({\cal Y}_k\) is wqo if and only if it contains none of the boundary properties. We show that the number of boundary properties is generally infinite. On the other hand, we prove that for each fixed k, there is a finite collection of boundary properties that allow to determine whether a class from \({\cal Y}_k\) is wqo or not.