---
abstract: 'If $G$ is a graph with vertex set $V$, let $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ as a configuration space of points on $G$, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ in the case where $G$ is a tree. Next, we use techniques of asymptotic algebra to prove statements about $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$, for general graphs $G$, whenever $n$ is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of [@R2].'
address: 'Department of Mathematics, University of Wisconsin - Madison.'
author:
- Eric Ramos
title: Configuration spaces of graphs with certain permitted collisions
---

Introduction
============

Introductory remarks
--------------------

Let $Y$ be a topological space. The **$n$-strand configuration space on $Y$** is the space $$\operatorname{Conf}_n(Y) := \{(y_1,\ldots,y_n) \in Y^n \mid y_i \neq y_j\}$$ For much of their history, configuration spaces have been studied in the case wherein $Y$ is a 