Completing and Extending Shellings of Vertex Decomposable Complexes
Completing and Extending Shellings of Vertex Decomposable Complexes
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DOI：
10.1137/21m1445119
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发表时间：
2020-11
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通讯作者：
Michaela Coleman;A. Dochtermann;Nathan Geist;Suho Oh
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作者：
Michaela Coleman;A. Dochtermann;Nathan Geist;Suho Oh
如果$\Delta$可以实现为$(n-1)$维单纯形的$d$-骨架的初始序列，则我们称$n$顶点上的纯$d$维单纯形$\Delta$是壳补的.Simon的一个著名猜想假设，任何可剥离的复合体都是剥离完备性的。在这篇注记中，我们证明了顶点可分解复形是壳层可完成的。事实上，我们证明了：如果$\Delta$是一个顶点可分解复形，则存在它的基集$V$的一个序，使得把$V$的最小缺失$(d+1)$-子集相加得到一个复形，它再次是顶点可分解的。我们探索了它在拟阵、移位复形以及$k-顶点可分解复形中的应用。我们还证明了，如果$\Delta$是至多$d+3$个顶点的$d$维复形，则可壳化、顶点可分解、壳化完备化和可延拓壳化的概念是等价的。
We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is 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, shifted complexes, as well as $k$-vertex decomposable complexes. 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.