Two robots moving geodesically on a tree

TL;DR

This paper investigates the geodesic complexity of configuration spaces of graphs, providing explicit geodesics and showing that geodesic complexity equals topological complexity for certain graph configurations.

Contribution

It determines the geodesic complexity for specific configuration spaces of trees and star graphs, establishing a link with topological complexity.

Findings

Geodesic complexity matches topological complexity for studied spaces.

Explicit geodesics are constructed for all pairs in the configuration spaces.

Results apply to both $ ext{l}_1$ and $ ext{l}_2$ metrics on graphs.

Abstract

We study the geodesic complexity of the ordered and unordered configuration spaces of graphs in both the $\ell_1$ and $\ell_2$ metrics. We determine the geodesic complexity of the ordered two-point $\varepsilon$-configuration space of any star graph in both the $\ell_1$ and $\ell_2$ metrics and of the unordered two-point configuration space of any tree in the $\ell_1$ metric, by finding explicit geodesics from any pair to any other pair, and arranging them into a minimal number of continuously-varying families. In each case the geodesic complexity matches the known value of the topological complexity.