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Properties of a q-Analogue of Zero Forcing

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Abstract

Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation of the game there were two options. Namely, to fill any one single vertex at the cost of a single token; or if any currently filled vertex has a unique non-filled neighbor, then the neighbor is filled for free. This paper investigates a q-analogue of zero forcing which introduces a third option involving an oracle. Basic properties of this game are established including determining all graphs which have minimal cost 1 or 2 for all possible q, and finding the zero forcing number for all trees when \(q=1\).

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Notes

  1. In terms of the linear algebraic philosophy of zero forcing, tokens were spent to probe and get additional information about the possible structure of the null space of a matrix and using the information to obtain a better bound on the nullity.

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Acknowledgements

The genesis of this paper was from the American Institute of Mathematics workshop Zero forcing and its applications, held in January 2017. We thank AIM for hosting their workshop and their support in bringing us together. Steve Butler was partially supported by a grant from the Simons Foundation (#427264). Shaun Fallat’s research is supported in part by an NSERC Discovery Research Grant, Application No.: RGPIN-2014-06036. Jephian C.-H. Lin was partially supported from the Young Scholar Fellowship Program by the Ministry of Science and Technology (MOST) in Taiwan, under Grant MOST108-2636-M-110-001. Boting Yang’s research is supported in part by an NSERC Discovery Research Grant, Application No.: RGPIN-2018-06800.

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Correspondence to Jephian C.-H. Lin.

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The original version of this article was revised: The LaTeX control sequence \deg is interpreted differently in the authors' and the publisher’s LaTeX setting. The authors’ intention is deg, as the degree of a vertex on a graph; however, it becomes a circle, as the degree for temperature, in the publisher’s environment. As a consequence, the mistakes occur whenever the macro \deg is used.

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Butler, S., Erickson, C., Fallat, S. et al. Properties of a q-Analogue of Zero Forcing. Graphs and Combinatorics 36, 1401–1419 (2020). https://doi.org/10.1007/s00373-020-02208-2

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  • DOI: https://doi.org/10.1007/s00373-020-02208-2

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