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Compact Metric Spaces with Infinite Cop Number.

Agelos Georgakopoulos1

  • 1Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.

Discrete & Computational Geometry
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PubMed
Summary
This summary is machine-generated.

Researchers disproved conjectures about the game of Cops and Robbers in metric spaces. A new metric on the 3-sphere (S 3) demonstrates an infinite cop number, challenging previous assumptions in pursuit-evasion games.

Keywords:
Cops and robberGeodesic metric spacePursuit-evasion

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Area of Science:

  • Topology and Geometry
  • Game Theory
  • Metric Spaces

Background:

  • The game of Cops and Robber, traditionally played on graphs, was recently extended to metric spaces by Mohar, unifying various pursuit-evasion games.
  • Mohar conjectured that a finite number of cops can always win on compact geodesic metric spaces.
  • He further conjectured that the cop number is bounded by homology group ranks on simplicial pseudo-manifolds.

Purpose of the Study:

  • To investigate the validity of Mohar's conjectures regarding the game of Cops and Robber in metric spaces.
  • To explore the cop number on compact geodesic metric spaces and simplicial pseudo-manifolds.

Main Methods:

  • Construction of a novel metric on the 3-sphere (S 3).
  • Analysis of the cop number within this newly defined metric space.

Main Results:

  • The study disproves Mohar's conjectures by constructing a metric on S 3 that results in an infinite cop number.
  • This finding demonstrates that the cop number is not always finite on compact geodesic metric spaces.

Conclusions:

  • Mohar's conjectures regarding finite cop numbers and homology bounds are disproven.
  • The study opens new avenues of research by raising more questions than it settles in the field of pursuit-evasion games on metric spaces.