Graphs containing finite induced paths of unbounded length

Abstract

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 $G^{\prime }$ 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.

arXiv comment28 pages, 3 figures

References in corpus (1)

• Sur l'énumération de structures discrètes, une approche par la théorie des relations