# Finite rigid subgraphs of pants graphs

## TL;DR

This paper constructs a finite subgraph within the pants graph of a surface that uniquely determines embeddings into other pants graphs, extending previous results to surfaces of arbitrary genus.

## Contribution

It identifies a finite rigid subgraph of the pants graph for surfaces of any genus, generalizing prior genus-zero results.

## Key findings

- Constructed finite rigid subgraph for arbitrary genus surfaces.
- Proved any embedding of this subgraph into another pants graph is induced by a surface embedding.
- Extended rigidity results from genus-zero to higher genus surfaces.

## Abstract

Let $S_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We identify a finite rigid subgraph $X_{g,n}$ of the pants graph $\mathcal P (S_{g,n})$, that is, a subgraph with the property that any simplicial embedding of $X_{g,n}$ into any pants graph $\mathcal P (S_{g',n'})$ is induced by an embedding $S_{g,n}\to S_{g',n'}$. This extends results of the third author for the case of genus zero surfaces.