Abstract
We give a characterization of robber-win strategies for general pursuit-evasion games with one evader and any finite number of pursuers on a finite graph. We also give an algorithm that solves robber-win games.
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Notes
See [9].
While it is often assumed that the cop moves first, and each player chooses a vertex at the beginning of time, starting with the cop, we consider a variation of the game in which the initial conditions of both players are chosen by nature initially and the robber moves before the cop.
Note that because \(k \ge l\), many l combinations are possible and hence \(N^{l}(v)\). Because we will not need to distinguish among those different possible \(N^{l}(v)\), this definition will not lead to any ambiguity in the remaining part of this section.
Note that because each vertex x has at least \(n+1\) neighbors, it will have at least \(\left( n+1\right) ^{2}\) neighbors of neighbors, and hence, \( N_{2}^{n-k}(v)\) is well-defined.
Note that by “characterize,” we mean the existence of a function called a strategy that dictates a player’s action at each move. In this paper, we do not address the computational complexity of the strategy.
An element of \(V\times V\) will a possible position of the evader and that of the pursuer for the case of one cop.
References
Parsons, T.D.: Pursuit-evasion in a graph. Lect. Notes Math. 642, 426–441 (1978)
Quilliot, A.: Jeux de points fixes sur les graphes, These de 3ieme Cycle. Universite de Paris VI, pp. 131–145 (1978)
Nowakowski, R., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Math. 43, 235–239 (1983)
Baird, W., Bonato, A.: Meyniel conjecture on the cop number: a survey. Working paper (2013)
Frankl, P.: Cops and robbers in graphs with large girths and cayley graphs. Discrete Appl. Math. 17, 301–305 (1987)
Wagner, Z.A.: Cops and robbers on diameter 2 graphs, Working paper (2014)
Aigner, M., Fromme, A.: A game of cops and robbers. Discrete Appl. Math. 8(1), 1–12 (1984)
Bonato, A., Pralat, P., Wang, C.: Pursuit-evasion in models of complex networks. Internet Math. 4, 419–436 (2009)
Berge, C.: Graphs. Elsevier Science Pub. Co., Amsterdam (1985)
Acknowledgements
We thank the Editor for his patience. We are grateful to two anonymous referees and the Associate Editor for their comments and suggestions on an earlier version of this paper. Other versions of this paper were presented at the 16th International Symposium on Dynamic Games and Applications held in Amsterdam, the Workshop on Games and Dynamics held in Chengdu, the 10th International ISDG Workshop held in Glasgow and the 15th SAET Conference, held in Cambridge. We would like to thank Tamer Basar, Yi-Chun Chen, Mahmoud Elchamie, John Howard, Jens Leth Hougaard, Sergey Kumkov, Takashi Kunimoto, Stephane Le Menec, Xiao Luo, Leon Petrosyan, and Roberto Serrano as well as other participants of the above conferences for their comments and suggestions.
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Communicated by Anita Schöbel.
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Ibragimov, G., Luckraz, S. On a Characterization of Evasion Strategies for Pursuit-Evasion Games on Graphs. J Optim Theory Appl 175, 590–596 (2017). https://doi.org/10.1007/s10957-017-1155-7
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DOI: https://doi.org/10.1007/s10957-017-1155-7