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Ambush Cops and Robbers

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Abstract

A variation of the Cops and Robber game is introduced in which the robber side consists of two robbers. The cops win by moving onto the same vertex as one of the robbers after a finite number of moves. As in the original game, the robber side can win by avoiding capture indefinitely. In this version, however, the robbers can also win by both moving onto the same vertex as the cop at the same time. Otherwise, the robbers must be located on distinct vertices. We present structural properties of ambush-copwin graphs, those graphs on which a single cop can guarantee a win. As well, we characterize ambush-copwin graphs of girth \(g \ge 4\) and classes of ambush-copwin graphs of girth 3, particularly a class of chordal graphs.

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References

  1. Bonato A., et al.: The Robber Strikes Back. In: Proceedings of ICC3, Adv. Intell. Syst. Comput., vol 24, pp. 3–12. Springer (2014)

  2. Blair, J.R.S., Peyton, B.: An Introduction to Chordal Graphs and Clique Trees. In: IMA Volumes in Mathematics and its Applications, vol. 56, pp. 1–29. Springer, New York (1993)

  3. Brandstädt, A., Le a, V.B., Spinrad,J.P.: Graph Classes: A Survey, SIAM Monogr. Discrete Math. Appl., Philadelphia, PA (1999)

  4. Chepoi, V.: On distance-preserving and domination elimination orderings. SIAM J. Discrete Math. 11(3), 414–436 (1998)

    Article  MathSciNet  Google Scholar 

  5. Clarke, N.E., MacGillivray, G.: Characterizations of \(k\)-copwin Graphs. Discrete Math. 312, 1421–1425 (2012)

    Article  MathSciNet  Google Scholar 

  6. Clarke, N.E., Nowakowski, R.J.: Cops, Robber, and Traps. Utilitas Math. 60, 91–98 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Clarke, N.E., Nowakowski, R.J.: Cops, Robber, and Photo Radar. Ars Combin. 56, 97–103 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Dirac, G.A.: On Rigid Circuit Graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)

    Article  MathSciNet  Google Scholar 

  9. Finbow , S., Fitzpatrick, S.L.: Cops and Scared Robber, preprint

  10. Fulkerson, D.R., Gross, O.A.: Incidence Matrices and Interval Graphs. Pac. J. Math. 15, 835–855 (1965)

    Article  MathSciNet  Google Scholar 

  11. Nowakowski, R.J., Winkler, P.: Vertex to Vertex Pursuit in a Graph. Discrete Math. 43, 23–29 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Quilliot, A., Jeux et Points Fixes sur les graphes, Thèse de 3ème cycle, Université de Paris VI, 1978, 131–145

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada

    Nancy E. Clarke, Melissa Creighton & Patrick Murray

  2. Department of Mathematics, Kwantlen Polytechnic University, Surrey, BC, Canada

    Asiyeh Sanaei

Authors
  1. Nancy E. Clarke
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  2. Melissa Creighton
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  3. Patrick Murray
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  4. Asiyeh Sanaei
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Correspondence to Nancy E. Clarke.

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Funding

N.E. Clarke acknowledges support from the NSERC (261518–2010); A. Sanaei acknowledges support from the Harrison McCain Foundation (through Acadia University).

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Clarke, N.E., Creighton, M., Murray, P. et al. Ambush Cops and Robbers. Graphs and Combinatorics 37, 2439–2457 (2021). https://doi.org/10.1007/s00373-021-02395-6

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  • DOI: https://doi.org/10.1007/s00373-021-02395-6

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