Configuration spaces of thick particles on a metric graph

Abstract

We study the topology of configuration spaces $F_r\left(\Gamma ,2\right)$ of two thick particles (robots) of radius $r>0$ moving on a metric graph $\Gamma$. As the size of the robots increases, the topology of $F_r\left(\Gamma ,2\right)$ varies. Given $\Gamma$ and $r$, we provide an algorithm for computing the number of path components of $F_r\left(\Gamma ,2\right)$. Using our main tool of PL Morse–Bott theory, we show that there are finitely many critical values of $r$ where the homotopy type of $F_r\left(\Gamma ,2\right)$ changes. We study the transition across a critical value $R\in \left(a,b\right)$ by computing the ranks of the relative homology groups of $\left(F_a\left(\Gamma ,2\right),F_b\left(\Gamma ,2\right)\right)$.