An accessibility theorem for infinite graph minors

In his recent collection [ 2 ] of open problems about infinite graphs, Halin defines the following hierarchy of graphs. He defines Γ0 to be the class of all graphs (finite or infinite), and for each ordinal μ > 0 he defines Γμ as the class of all graphs containing, for each λ < μ, infinitely many disjoint connected graphs from Γλ. Thus, Γ1 is the class of infinite graphs, Γ2 is the class of all graphs containing infinitely many disjoint connected infinite graphs (or equivalently, either infinitely many disjoint rays or infinitely many disjoint infinite stars), and so on. Halin then asks which graphs, if any, lie in every Γμ. He notes that Γλ ⊇ Γμ whenever λ < μ, and that any graph with a minor in Γμ is itself in Γμ (induction on μ). We might add the observation that the sequence of Γμs does not become stationary, i.e. that for every ordinal μ there is a (connected) graph in Γμ Γμ+1; this, too, follows easily by induction on μ. Intuitively, it may be natural to consider not the graph properties Γμ themselves but their complements Γμ. As observed above, these classes are closed under taking minors, and they form an increasing sequence Γ0 ⊆ Γ1 ⊆ . . . . The question then arises whether every graph is captured by this sequence, and if not then which graphs are. Following Halin, let us define as the order o(G) of a graph G the least ordinal μ (if one exists) such that G / ∈ Γμ. Note that if G has an order and H is a minor of G, then H too has an order and o(H) o(G). Which graphs, then, have an order? At first glance, this may look like a difficult question. Indeed, it even seems unclear whether or not every graph has an order. As Halin observes, however, there is a simple sufficent condition for not having an order: if a connected graph G contains infinitely many disjoint copies of itself, then induction on μ shows at once that G ∈ Γμ for all ordinals μ.