Volume 22, issue 2 (2022)

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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

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

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geodesic, configuration space, topological robotics, graphs
Mathematical Subject Classification
Primary: 53C22, 55M30, 55R80
Secondary: 68T40
Received: 2 July 2020
Revised: 8 December 2020
Accepted: 11 January 2021
Published: 3 August 2022
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