Computing a configuration skeleton for motion planning of two round robots on a metric graph

A connected metric graph G with n vertices and without loops and multiple edges is given as an n × n-matrix whose entry a_ij is the length of a single edge between vertices i ≠ j. A robot in the metric graph G is the metric ball with a center x ϵ G and a radius r > 0. The configuration space OC(G, r) of 2 ordered robots in G is the set of all centers (x, y)ϵ G×G such that x, y are at least 2r away from each other. We introduce the configuration skeleton CS(G, r) ⊂ OC(G, r) that captures all connectivity information of the larger space OC(G, r). We design an algorithm of time complexity O(n²) to find all connected components of OC(G, r) that are maximal subsets of all safe positions (x, y) connectable by collision-free motions of the two round robots.