Finite subgraphs of an extension graph

TL;DR

The paper introduces a method to generate finite subgraphs of an extension graph from a sequence of induced subgraphs, showing all such subgraphs can be obtained through this process.

Contribution

It provides a new inductive construction for finite subgraphs of extension graphs using doubling along stars, establishing a comprehensive characterization.

Findings

Every finite induced subgraph of the extension graph is contained in some $\

The sequence $\\{\\\Gamma_i\\\}$ effectively captures all finite subgraphs.

The method applies to any finite graph $\\Gamma$ and its extension graph.

Abstract

Let $\Gamma$ be a finite graph and let $\Gamma^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{\Gamma_i\}$ of finite induced subgraphs of $\Gamma^{\mathrm{e}}$ through successive applications of an operation called "doubling along a star". Then we show that every finite induced subgraph of $\Gamma^{\mathrm{e}}$ is isomorphic to an induced subgraph of some $\Gamma_i$.