The age $\mathcal{A}(G)$ 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 \emph{path-minimal} if it contains finite induced paths of unbounded length and every induced subgraph $G'$ with this property embeds $G$. We construct $2^{\aleph_0}$ 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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• 05C38 - Paths and cycles
• 05C60 - Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
• 05C78 - Graph labelling (graceful graphs, bandwidth, etc.)
• 06A07 - Combinatorics of partially ordered sets
• 68R15 - Combinatorics on words