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Shellability is NP-Complete

Authors Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner

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Xavier Goaoc
Pavel Paták
Zuzana Patáková
Martin Tancer
Uli Wagner

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Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. Shellability is NP-Complete. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 41:1-41:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


We prove that for every d >= 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d >= 2 and k >= 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d >= 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.
  • Shellability
  • simplicial complexes
  • NP-completeness
  • collapsibility


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