Topological complexity of configuration spaces of fully articulated graphs and banana graphs

TL;DR

This paper computes the topological complexity of configuration spaces for non-tree graphs, specifically fully articulated and banana graphs, revealing differences between ordered and unordered cases and extending previous results.

Contribution

It extends the computation of topological complexity to non-tree graphs and shows unordered configuration spaces can have lower complexity than ordered ones.

Findings

Topological complexity computed for fully articulated and banana graphs.

Unordered configuration spaces can have lower topological complexity than ordered ones.

Extended the known results for trees to more general graph classes.

Abstract

In this paper we determine the topological complexity of configuration spaces of graphs which are not necessarily trees, which is a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs. We also complete the computation in the case of trees to include configuration spaces with any number of points, extending a proof of Farber. At the end we show that an unordered configuration space on a graph does not always have the same topological complexity as the corresponding ordered configuration space (not even when they are both connected). Surprisingly, in our counterexamples the topological complexity of the unordered configuration space is in fact smaller than for the ordered one.