Cops and Robbers on Dynamic Graphs: Offline and Online Case

We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the underlying graph of the model. The cops and the robber take their turns on the current graph in...

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Published in:Discrete mathematics and theoretical computer science Vol. 25:1; no. Discrete Algorithms; pp. 1 - 20
Main Authors: Balev, Stefan, Jiménez Laredo, Juan, Lamprou, Ioannis, Pigné, Yoann, Sanlaville, Eric
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.01.2023
Discrete Mathematics & Theoretical Computer Science
Subjects:
[info.info-dm]computer science [cs]/discrete mathematics [cs.dm]
[info.info-ro]computer science [cs]/operations research [cs.ro]
Algorithms
Apexes
Computer Science
cops and robbers
Discrete Mathematics
dynamic graphs
Evolution
Games
Graph theory
Graphs
offline
online
Operations Research
Robbery
sparse
cops and robbers
dynamic graphs
offline
online
sparse
ISSN:1365-8050, 1462-7264, 1365-8050
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Abstract We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the underlying graph of the model. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n 2k+1 T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs, i.e., with at most linear number of edges. We prove λ + 1 cops suffice to capture the robber in any underlying graph with n − 1 + λ edges. Further, we define a family of underlying graphs with n−1+λ edges where λ−1 cops are necessary (and sufficient) for capture.
AbstractList We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the underlying graph of the model. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n 2k+1 T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs, i.e., with at most linear number of edges. We prove λ + 1 cops suffice to capture the robber in any underlying graph with n − 1 + λ edges. Further, we define a family of underlying graphs with n−1+λ edges where λ−1 cops are necessary (and sufficient) for capture.
We examine the classic game of Cops and Robbers played on dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the (static) underlying graph. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n2k+1T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs with at most a linear number of edges. We prove A + 1 cops suffice to capture the robber in any underlying graph with n - 1 + A edges. Furthermore, we define a family of underlying graphs with n - 1 + A edges where A - 1 cops are necessary (and sufficient) for capture.
Author Pigné, Yoann
Lamprou, Ioannis
Balev, Stefan
Jiménez Laredo, Juan
Sanlaville, Eric
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  surname: Sanlaville
  fullname: Sanlaville, Eric
  organization: Equipe Réseaux d'interactions et Intelligence Collective
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Snippet We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a...
We examine the classic game of Cops and Robbers played on dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph...
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SubjectTerms [info.info-dm]computer science [cs]/discrete mathematics [cs.dm]
[info.info-ro]computer science [cs]/operations research [cs.ro]
Algorithms
Apexes
Computer Science
cops and robbers
Discrete Mathematics
dynamic graphs
Evolution
Games
Graph theory
Graphs
offline
online
Operations Research
Robbery
sparse
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