Graphs containing finite induced paths of unbounded length
Abstract
The age A(G) of a graph G (undirected and without loops) is the collection of finite induced subgraphs of G, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is path-minimal if it contains finite induced paths of unbounded length and every induced subgraph G' with this property embeds G. We construct 20 path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs.
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