We study the topology of configuration spaces of two thick particles (robots) of radius moving on a metric graph . As the size of the robots increases, the topology of varies. Given and , we provide an algorithm for computing the number of path components of . Using our main tool of PL Morse–Bott theory, we show that there are finitely many critical values of where the homotopy type of changes. We study the transition across a critical value by computing the ranks of the relative homology groups of .
"Configuration spaces of thick particles on a metric graph." Algebr. Geom. Topol. 11 (4) 1861 - 1892, 2011. https://doi.org/10.2140/agt.2011.11.1861