# Topological complexity of unordered configuration spaces of certain graphs

## TL;DR

This paper investigates the topological complexity of unordered configuration spaces of points on graphs, providing combinatorial methods to compute it for various classes of graphs and relating it to graph properties.

## Contribution

It introduces a combinatorial approach to compute the topological complexity of discretized configuration spaces on graphs, with explicit results for specific graph classes.

## Key findings

- For large classes of graphs, TC(UD^n(Γ))=2m(Γ)+1 when robots ≥ 2m(Γ).
- When robots ≤ half the number of vertex-disjoint cycles, TC(UD^n(Γ))=2n+1.
- Provides formulas linking topological complexity to graph properties like degree and cycles.

## Abstract

The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted here by $UC^n(\Gamma),$ can be viewed as the space of all configurations of $n$ unlabeled robots on a system of one-dimensional tracks, which is interpreted as a graph $\Gamma.$ The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by $m(\Gamma).$ We discuss a combinatorial approach to compute the topological complexity of a "discretized" version of this space, $UD^n(\Gamma),$ and give results for certain classes of graphs. In the first case, we show that for a large class of graphs, as long as the number of robots is at least $2m(\Gamma)$, then $TC(UD^n(\Gamma))=2m(\Gamma)+1.$ In the second, we show that as long as the number of robots is at most half the number of vertex-disjoint cycles in $\Gamma,$ we have $TC(UD^n(\Gamma))=2n+1.$