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Séminaire Lotharingien de Combinatoire, 85B.88 (2021), 12 pp.

# Michaela Coleman, Anton Dochtermann, Nathan Geist and Suho Oh

# Completing and Extending Shellings
of Vertex Decomposable Complexes

**Abstract.**
We say that a pure *d*-dimensional simplicial complex Δ on *n* vertices is *shelling completable* if Δ can be realized as the initial sequence of some shelling of Δ_{n-1}^{(d)}, the *d*-skeleton of the (*n*-1)-dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable. We prove that vertex decomposable complexes are shelling completable. In fact we show that if Δ is a vertex decomposable complex then there exists an ordering of its ground set *V* such that adding the revlex smallest missing (*d*+1)-subset of *V* results in a complex that is again vertex decomposable. We explore applications to matroids, shifted complexes, as well as *k*-vertex decomposable complexes. We also show that if Δ is a *d*-dimensional complex on at most *d*+3 vertices then the notions of shellable, vertex decomposable, shelling completable, and extendably shellable are all equivalent.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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