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We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$vertices is \emph{shelling completable} if $\Delta$ can be realized as theinitial sequence of some shelling of $\Delta_{n-1}^{(d)}$, the $d$-skeleton ofthe $(n-1)$-dimensional simplex. A well-known conjecture of Simon posits thatany shellable complex is shelling completable. In this note we prove thatvertex decomposable complexes are shelling completable. In fact we show that if$\Delta$ is a vertex decomposable complex then there exists an ordering of itsground set $V$ such that adding the revlex smallest missing $(d+1)$-subset of$V$ results in a complex that is again vertex decomposable. We exploreapplications to matroids and shifted complexes, as well as connections toridge-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 ofshellable, vertex decomposable, shelling completable, and extendably shellableare all equivalent.

Completing and extending shellings of vertex decomposable complexes