# Exhausting pants graphs of punctured spheres by finite rigid sets

## TL;DR

The paper constructs an increasing sequence of finite rigid sets in the pants graph of an n-punctured sphere that exhausts the entire graph, providing new insights into its structure.

## Contribution

It introduces a method to build an increasing sequence of finite rigid sets that cover the entire pants graph of punctured spheres.

## Key findings

- Sequence of finite rigid sets exhausts the pants graph
- Provides a new approach to understanding the structure of pants graphs
- Establishes a foundation for further topological and geometric analysis

## Abstract

Let $S_{0,n}$ be an $n$-punctured sphere. For $n\geq 4$, we construct a sequence $(\mathcal{X}_i)_{i\in\mathbb{N}}$ of finite rigid sets in the pants graph $\mathcal{P}(S_{0,n})$ such that $\mathcal{X}_1 \subset \mathcal{X}_2 \subset ...\subset\mathcal{P}(S_{0,n})$ and $\bigcup_{i\geq1}\mathcal{X}_i=\mathcal{P}(S_{0,n})$.