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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Vydáno v:Discrete mathematics and theoretical computer science Ročník 25:1; číslo Discrete Algorithms; s. 1 - 20
Hlavní autoři: Balev, Stefan, Jiménez Laredo, Juan, Lamprou, Ioannis, Pigné, Yoann, Sanlaville, Eric
Médium: Journal Article
Jazyk:angličtina
Vydáno: Nancy DMTCS 01.01.2023
Discrete Mathematics & Theoretical Computer Science
Témata:
[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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Popis
Shrnutí: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.
Bibliografie:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.8784