论文标题
简单复合物和非覆盖功能复合物的可折叠性数量的界限
Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs
论文作者
论文摘要
韦格纳（Wegner）引入了简单复合物的可折叠性数量，以了解凸集的相交模式。该数字在多种Helly型结果中也起着重要作用。我们表明，超图$ \ MATHCAL {H} $是$ | V（\ Mathcal {h）} |-γ_I（\ Mathcal {h}） - 1 $ -Collapsible，其中$γ_i（\ Mathcal {h}）$是独立性Domination Domination Domination Domination formeprach。这将Choi，Kim和Park的结果从图表扩展到了超图。此外，可以作为我们结果的特殊情况获得，在Kim和Kim为非覆盖复合物的Leray数字上给出的强大独立支配数字而言。 通常，在复合物的可折叠率及其众所周知的上限之间可能存在很大的差距。 在本文中，我们构建了一个上限$ \ MATHCAL {M} _K（x）$的序列，用于简单复杂的$ x $的可折叠率，该数字位于此间隙中。我们还表明，如果基础复合物为$ k $ - vertex doposoposable，则由$ \ Mathcal {M} _K $给出的绑定很紧。
The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number also plays an important role in a variety of Helly type results. We show that the non-cover complex of a hypergraph $\mathcal{H}$ is $|V(\mathcal{H)}|- γ_i(\mathcal{H})-1$-collapsible, where $γ_i(\mathcal{H})$ is the generalization of independence domination number of a graph to hypergraph. This extends the result of Choi, Kim and Park from graphs to hypergraphs. Moreover, the upper bound in terms of strong independence domination number given by Kim and Kim for the Leray number of the non-cover complex of a hypergraph can be obtained as a special case of our result. In general, there can be a large gap between the collapsibility number of a complex and its well-known upper bounds. In this article, we construct a sequence of upper bounds $\mathcal{M}_k(X)$ for the collapsibility number of a simplicial complex $X$, which lie in this gap. We also show that the bound given by $\mathcal{M}_k$ is tight if the underlying complex is $k$-vertex decomposable.
