Completing and extending shellings of vertex decomposable complexes

TL;DR

This paper proves that vertex decomposable complexes are shelling completable, advancing understanding of shellability in simplicial complexes and exploring implications for matroids, shifted complexes, and related structures.

Contribution

It establishes that vertex decomposable complexes can be extended to shellings, confirming a special case of Simon’s conjecture and providing new methods for constructing shellings.

Findings

Vertex decomposable complexes are shelling completable.

For complexes with at most d+3 vertices, shellable, vertex decomposable, and shelling completable are equivalent.

Adding the revlex smallest missing face preserves vertex decomposability.

Abstract

We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is \emph{shelling completable} if $\Delta$ can be realized as the initial sequence of some shelling of $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable. In this note we prove that vertex decomposable complexes are shelling completable. In fact we show that if $\Delta$ 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 and shifted complexes, as well as connections to ridge-chordal complexes and $k$-decomposability. We also show that if $\Delta$ 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.