Two robots moving geodesically on a tree

Davis, Donald M; Harrison, Michael; Recio-Mitter, David

doi:10.2140/agt.2022.22.785

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Two robots moving geodesically on a tree

Donald M Davis, Michael Harrison and David Recio-Mitter

Algebraic & Geometric Topology 22 (2022) 785–814

DOI: 10.2140/agt.2022.22.785

Abstract

We study the geodesic complexity of the ordered and unordered configuration spaces of graphs in both the $ℓ_{1}$ and $ℓ_{2}$ metrics. We determine the geodesic complexity of the ordered two-point $𝜀$ –configuration space of any star graph in both the $ℓ_{1}$ and $ℓ_{2}$ metrics and of the unordered two-point configuration space of any tree in the $ℓ_{1}$ 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.

Keywords

geodesic, configuration space, topological robotics, graphs

Mathematical Subject Classification

Primary: 53C22, 55M30, 55R80

Secondary: 68T40

References

Publication

Received: 2 July 2020

Revised: 8 December 2020

Accepted: 11 January 2021

Published: 3 August 2022

Authors

Donald M Davis
Department of Mathematics Lehigh University Bethlehem, PA United States

Michael Harrison
Institute for Advanced Study Princeton, NJ United States

David Recio-Mitter
Department of Mathematics Lehigh University Bethlehem, PA United States

Volume 22, issue 2 (2022)