Graphs containing finite induced paths of unbounded length

The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is thecollection of finite induced subgraphs of $G$, considered up to isomorphy andordered by embeddability. It is well-quasi-ordered (wqo) for this order if itcontains no infinite antichain. A graph is \emph{path-minimal} if it containsfinite induced paths of unbounded length and every induced subgraph $G'$ withthis property embeds $G$. We construct $2^{\aleph_0}$ path-minimal graphs whoseages are pairwise incomparable with set inclusion and which are wqo. Ourconstruction is based on uniformly recurrent sequences and lexicographical sumsof labelled graphs.Comment: 28 pages, 3 figure