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Shellings from Relative Shellings, with an Application to NP-Completeness

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Abstract

Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. Shellings of a complex expose a large amount of information in a helpful way, but are not easy to construct, often requiring deep information about the structure of the complex. It is natural to ask whether shellings may be efficiently found computationally. In a recent paper, Goaoc, Paták, Patáková, Tancer, and Wagner gave a negative answer to this question (assuming \(\mathsf {P}\ne \mathsf {NP}\)), showing that the problem of deciding whether a simplicial complex is shellable is \(\mathsf {NP}\)-complete. In this paper, we give simplified constructions of various gadgets used in the \(\mathsf {NP}\)-completeness proof of these authors. Using these gadgets combined with relative shellability and other ideas, we also exhibit a simpler proof of the \(\mathsf {NP}\)-completeness of the shellability decision problem. Our method systematically uses relative shellings to build up large shellable complexes with desired properties.

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Acknowledgements

We thank an anonymous referee for careful and thoughtful comments, which helped us improve readability. We also thank Abhishek Rathod for his helpful feedback on a draft of the paper.

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Correspondence to Russ Woodroofe.

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This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects J1-9108, N1-0160, and J1-2451)

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Santamaría-Galvis, A., Woodroofe, R. Shellings from Relative Shellings, with an Application to NP-Completeness. Discrete Comput Geom 66, 792–807 (2021). https://doi.org/10.1007/s00454-020-00273-1

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