Topological complexity of unordered configuration spaces of certain graphs

Abstract

The unordered configuration space of n points on a graph Γ, denoted here by UCn(Γ), 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 Γ. The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by m(Γ). We discuss a combinatorial approach to compute the topological complexity of a “discretized” version of this space, UDn(Γ), 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(Γ), then TC(UDn(Γ))=2m(Γ)+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 Γ, we have TC(UDn(Γ))=2n+1.