Graphs containing finite induced paths of unbounded length

TL;DR

This work constructs path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo, based on uniformly recurrent sequences and lexicographical sums of labelled graphs.

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.

Figures (2)

• Figure 1: Sum of two labeled paths. For $n\in {\mathbb N}$ set $\ell(2n)=0$ and $\ell(2n+1)=1$. The operation $\star$ is the Boolean sum.
• Figure 2: Two examples of paths not contained in one component of $\widehat{G}_{(u, \star)}$ if $\star$ is the Boolean sum. The thick vertical lines represent components of $\widehat{G}_{(u, \star)}$.

Theorems & Definitions (70)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• Theorem 5
• Lemma 6
• proof
• Theorem 7
• Theorem 8
• proof