## Shellability is NP-complete

**Authors:** Thinh Nguyen

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. Another
simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-
shellable.

**Comments:** 19 Pages.

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### Submission history

[v1] 2018-07-05 14:59:41

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