A Knaster--Reichbach type theorem for graph structures

TL;DR

This paper demonstrates that a generic finite graph object, topologically similar to a Cantor set, exhibits a property allowing local symmetries on certain closed, isolated vertex subsets to extend globally to the entire graph.

Contribution

It establishes a Knaster--Reichbach type extension property for a generic object in the category of finite graphs, linking topological and graph-theoretic symmetries.

Findings

Homeomorphisms extend to automorphisms of the entire graph.

Isomorphisms between nowhere dense closed sets extend globally.

The generic graph object has a Cantor set-like topological structure.

Abstract

We study the properties of a generic object $\mathbb{P}$ in the category of finite graphs. It turns out that this object, being topologically a Cantor set, has the Knaster--Reichbach type property. Namely, every homeomorphism and isomorphism $h\colon K\to L$ where $K$ and $L$ are nowhere dense closed sets in $\mathbb{P}$ and consisting only of isolated vertices in $K$ and $L$ can be extended to the autohomeomorphism and autoisomorphism of the whole graph $\mathbb{P}$.