Graphs containing finite induced paths of unbounded length

TL;DR

This paper constructs a large family of path-minimal graphs with complex age structures, demonstrating their properties using advanced combinatorial and order-theoretic methods.

Contribution

It introduces a method to construct uncountably many path-minimal graphs with incomparable ages that are well-quasi-ordered, expanding understanding of graph ages and their embeddings.

Findings

Constructed 2^{} path-minimal graphs with incomparable ages.

Demonstrated these graphs are well-quasi-ordered.

Used sequences and lexicographical sums for construction.

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 $G'$ 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.