# Shellability is NP-complete

Abstract : We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is \textup{NP}-hard, hence \textup{NP}-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional, simplicial complex is $k$-decomposable is \textup{NP}-hard. For $d \ge 3$, both problems remain \textup{NP}-hard when restricted to contractible pure $d$-dimensional complexes.
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Conference papers

https://hal-upec-upem.archives-ouvertes.fr/hal-01744101
Contributor : Xavier Goaoc <>
Submitted on : Tuesday, March 27, 2018 - 10:03:00 AM
Last modification on : Wednesday, February 6, 2019 - 10:22:02 AM

### Identifiers

• HAL Id : hal-01744101, version 1
• ARXIV : 1711.08436

### Citation

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner. Shellability is NP-complete. International Symposium on Computational Geometry, 2018, Budapest, Hungary. ⟨hal-01744101⟩

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