Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs
arxiv(2022)
摘要
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 ℋ is |V(ℋ)|-
γ_i(ℋ)-1-collapsible, where γ_i(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.
available in literature.
In general, there can be a large gap between the collapsibility number of a
complex and its well-known upper bounds.
number.
In this article, we construct a sequence of upper bounds 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 ℳ_k is tight if the
underlying complex is k-vertex decomposable.
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