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Two-player pebbling on diameter 2 graphs

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Abstract

A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root. The pebbling number of a graph \(\pi (G)\) is the minimum number of pebbles needed so every distribution of \(\pi (G)\) pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations of various classes of graphs for which each player has a winning strategy. We will characterize the winning player for a specific class of diameter two graphs.

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References

  • Albert MH, Nowakowski RJ, Wolfe D (2019) Lessons in play: an introduction to combinatorial game theory. CRC Press, Boca Raton

    Book  Google Scholar 

  • Chung FRK (1989) Pebbling in hypercubes. SIAM J Discrete Math 2(4):467–472

    Article  Google Scholar 

  • Conway JH, Guy RK, Berlekamp ER (2003) Winning ways for your mathematical plays

  • Faigle U, Kern W, Kierstead H, Trotter WT (1991) On the game chromatic number of some classes of graphs

  • Fisher M, Tennenhouse C (2017) The game of blocking pebbles. arXiv preprint. arXiv:1712.01173

  • Hurlbert G (2013) General graph pebbling. Discrete Appl Math 161(9):1221–1231

    Article  Google Scholar 

  • Hurlbert GH (1999) A survey of graph pebbling. In: Proceedings of the thirtieth southeastern international conference on combinatorics, Graph theory, and computing (Boca Raton, FL, 1999), vol 139, pp 41–64

  • Krivelevich M, Kronenberg G (2015) Random-player maker-breaker games. arXiv preprint. arXiv:1502.00445

  • Lemke P, Kleitman D (1989) An addition theorem on the integers modulo \(n\). J Number Theory 31(3):335–345

    Article  Google Scholar 

  • Pachter L, Snevily HS, Voxman B (1995) On pebbling graphs. Congr Numer 107:65–80

    Google Scholar 

  • Prudente M (2015) Two-player variation on graph pebbling. Thesis (Ph.D.), Lehigh University

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Authors and Affiliations

  1. Department of Mathematics, Lehigh University, Bethlehem, PA, USA

    Garth Isaak

  2. Department of Science and Mathematics, Alvernia University, Reading, PA, USA

    Matthew Prudente

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  1. Garth Isaak
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  2. Matthew Prudente
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Correspondence to Matthew Prudente.

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Isaak, G., Prudente, M. Two-player pebbling on diameter 2 graphs. Int J Game Theory 50, 581–596 (2021). https://doi.org/10.1007/s00182-021-00766-0

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  • DOI: https://doi.org/10.1007/s00182-021-00766-0

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