A Note On Immersion Intertwines Of Infinite Graphs

TL;DR

This paper constructs specific infinite graphs to demonstrate that the class of infinite graphs ordered by immersion lacks the finite intertwine property, revealing limitations in the structure of infinite graph immersions.

Contribution

It introduces a novel construction of infinite graphs and an antichain set to show the non-existence of the finite intertwine property in infinite graph immersions.

Findings

Constructed infinite graphs $G_1$ and $G_2$ with specific immersion properties

Established an infinite antichain of graphs with particular subgraph relations

Proved the class of infinite graphs by immersion does not have the finite intertwine property

Abstract

We present a construction of two infinite graphs $G_1$ and $G_2$, and of an infinite set $\mathscr{F}$ of graphs such that $\mathscr{F}$ is an antichain with respect to the immersion relation and, for each graph $G$ in $\mathscr{F}$, both $G_1$ and $G_2$ are subgraphs of $G$, but no graph properly immersed in $G$ admits an immersion of $G_1$ and of $G_2$. This shows that the class of infinite graphs ordered by the immersion relation does not have the finite intertwine property.