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On Properties of Pebble Assignment Graphs

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Abstract

A pebble assignment \((S_G)\) on a finite simple graph G is a distribution of a finite number of pebbles on the vertices of G. A standard pebbling move consists of removing two pebbles from a vertex x of G while adding one pebble to a vertex adjacent to x. In this paper, we introduce the new notion of a pebble assignment graph. Formally, this is a Hasse diagram \([S_{G}]\) whose nodes correspond to all assignments that can be reached from \((S_G)\) by applying a sequence of pebbling moves. Edges of \([S_G]\) adjoin any two assignments where one can be reached from the other via a single pebbling move (i.e., from “parent" to “child"). We investigate properties of this new class of graphs, addressing such questions as which abstract graphs can be realized as pebbling assignment graphs, and for which graphs G does there exist an assignment \((S_G)\) such that \([S_G]\cong G\)?

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

The authors wish to thank the referees for their careful reading of the manuscript and valuable comments.

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The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    M. Lind

  2. DIMACS-Rutgers University, Piscataway, NJ, 08854, USA

    E. Fiorini

  3. Villanova University, Villanova, PA, 19085, USA

    A. Woldar

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  1. M. Lind
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  2. E. Fiorini
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  3. A. Woldar
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Contributions

Authors contributed to the study conception and design. Material preparation and analysis were performed by Max Lind, Eugene Fiorini and Andrew Woldar. The first draft of the manuscript was written by Andrew Woldar and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to E. Fiorini.

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Lind, M., Fiorini, E. & Woldar, A. On Properties of Pebble Assignment Graphs. Graphs and Combinatorics 38, 45 (2022). https://doi.org/10.1007/s00373-021-02453-z

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  • DOI: https://doi.org/10.1007/s00373-021-02453-z

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