Abstract
Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal \(\alpha \), any winning strategy has complexity at least \(0^{(\alpha )}\). Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are explored. The Turing computability of a binary relation used to classify cop-win graphs is studied, and the computational difficulty of determining the winner for locally finite computable graphs is discussed.
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Stahl, R.D. Computability and the game of cops and robbers on graphs. Arch. Math. Logic 61, 373–397 (2022). https://doi.org/10.1007/s00153-021-00794-3
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DOI: https://doi.org/10.1007/s00153-021-00794-3
Keywords
- Cops and robbers on graphs
- Infinite graphs
- Computability
- Effective strategy
- Games on graphs
- Locally finite trees