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On the Conjecture of the Smallest 3-Cop-Win Planar Graph

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Theory and Applications of Models of Computation (TAMC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10185))

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Abstract

In the game of Cops and Robbers on a graph \(G=(V,E)\), k cops try to catch a robber. The minimum number of cops required to win is called the cop number, denoted by c(G). For a planar graph G, it is known that \(c(G)\le 3\). It is a conjecture that the regular dodecahedral graph of order 20 is the smallest planar graph whose cop number is three. As the very first attack on this conjecture, we provide the following evidences in this paper: (1) any planar graph of order at most 19 has the winning vertex at which two cops can capture the robber, and (2) a special planar graph of order 19 that is constructed from the regular dodecahedral graph has the cop number of two.

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Notes

  1. 1.

    Probably, the cycle of length at least five does not exist at all when \(\delta (G)=5\).

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Authors and Affiliations

  1. School of Science and Technology, Tokai University, 4-1-1 Kitakaname, Hiratsuka, 259-1292, Japan

    Photchchara Pisantechakool

  2. School of Information Science and Technology, Tokai University, 4-1-1 Kitakaname, Hiratsuka, 259-1292, Japan

    Xuehou Tan

Authors
  1. Photchchara Pisantechakool
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  2. Xuehou Tan
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Corresponding author

Correspondence to Photchchara Pisantechakool .

Editor information

Editors and Affiliations

  1. Anna University, Chennai, India

    T.V. Gopal

  2. Universität Bern , Bern, Switzerland

    Gerhard Jäger

  3. Universität Bern , Bern, Switzerland

    Silvia Steila

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Pisantechakool, P., Tan, X. (2017). On the Conjecture of the Smallest 3-Cop-Win Planar Graph. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_36

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  • DOI: https://doi.org/10.1007/978-3-319-55911-7_36

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55910-0

  • Online ISBN: 978-3-319-55911-7

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