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The Explorer–Director Game on Graphs

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Abstract

The Explorer-Director game, first introduced by Nedev and Muthukrishnan, can be described as a game where two players—Explorer and Director—determine the movement of a token that is positioned on the vertices of some given graph. At each time step, the Explorer specifies a distance that the token must move with an aim to maximize the total number of vertices ultimately visited. However, the Director adversarially chooses where to move token in an effort to minimize this number. The game ends when no new vertices can be visited. Given a graph G and a starting vertex v, the number of vertices that are visited under optimal play is denoted by \(f_d(G,v)\). In this paper, we first reduce the study of \(f_d (G,v)\) to the determination of the minimum sets of vertices that are closed in a certain combinatorial sense, thus providing a structural understanding of each player’s optimal strategies. As an application, we provide some exact results as well as more general bounds when G is a square lattice or tree. In the case of trees, we also provide a complete solution even in the more restrictive setting where the strategy used by the Explorer is not allowed to depend on their opponent’s responses. In addition to this paper, a supplementary companion note will be posted to arXiv providing additional results about the game in a variety of specific graph families.

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Data sharing not applicable to this article as no datasets were generated or analyzed.

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Acknowledgements

The authors would like to thank the reviewers for their careful reading. The authors would like to thank the organizers and participants of the Polymath Research Experience for Undergraduates where this work was conducted.

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Authors and Affiliations

  1. Department of Mathematics, Swarthmore College, Swarthmore, PA, USA

    Pat Devlin

  2. School of Computing, Queens University, Kingston, ON, Canada

    Erin Meger

  3. Department of Mathematics, Cooper Union, New York, NY, USA

    Abigail Raz

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  1. Pat Devlin
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  2. Erin Meger
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  3. Abigail Raz
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Correspondence to Abigail Raz.

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Polymath REU Participants: Ben Bates, Tyler Beauregard, Daniel Brey, Elaine Danielson, Mayur Lahane, Angela Li, Mohamed Lotfi, Stephanie Marsh, Oskar Mathiasen, Audrey McMillion, Dhruva Mehta, Joseph Mescall, Jonah Nan, Jarod Palubiski, Cac Phan, Isabel Sacksteder, Danielle Solomon, João Tavares, Lincoln Updike, Shannon Vogel, James Wash, Charles Weng, Erica Zhang, Mengxue Zhang.

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Devlin, P., Meger, E., Raz, A. et al. The Explorer–Director Game on Graphs. Graphs and Combinatorics 39, 48 (2023). https://doi.org/10.1007/s00373-023-02638-8

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  • DOI: https://doi.org/10.1007/s00373-023-02638-8

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