Grouped Stirling complexes

TL;DR

This paper introduces Grouped Stirling complexes, a new type of configuration space for grouped robots on graphs, analyzing their connectivity and cell structure under specific constraints.

Contribution

It defines Grouped Stirling complexes and establishes their path-connectedness for three or more groups, along with cell count formulas in particular cases.

Findings

$S_{\vec r}(G)$ is path-connected if there are at least three groups.

The paper provides formulas for the number of cells in $S_{\vec r}(G)$ in certain scenarios.

The space has a closed cell structure, facilitating topological analysis.

Abstract

Given a graph $G$, a configuration space of $G$ can be thought of as the set of all possible configurations of "robots" which can move throughout $G$, subject to some constraints. We introduce a type of configuration space which we call Grouped Stirling complexes, denoted by $S_{\vec r}(G)$, in which we place robots in groups subject to two constraints. First, there must be at least one robot on each vertex of $G$, and second, any two robots from the same group must be "separated by at least one full open edge" of $G$. The space $S_{\vec r}(G)$ has a closed cell structure, which means it can be built out of cells of various dimensions. Our main results show $S_{\vec r}(G)$ is path-connected, provided there are at least three groups, and determine the number of cells of $S_{\vec r}(G)$ in certain cases.