Collapsibility of Non-Cover Complexes of Graphs

Choi, Ilkyoo; Kim, Jinha; Park, Boram

doi:10.37236/8684

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…In this section, we prove the second part of Theorem 1.6. When H is a graph, it is implied by the main result in [7]. Our result gives an alternative proof with a slight extension.…”

Section: Strong Independence Domination Numbersmentioning

confidence: 66%

Domination numbers and noncover complexes of hypergraphs

Kim

2021

Preprint

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Let H be a hypergraph on a finite set V . A cover of H is a set of vertices that meets all edges of H. If W is not a cover of H, then W is said to be a noncover of H. The noncover complex of H is the abstract simplicial complex whose faces are the noncovers of H. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called tight paths and tight cycles. This extends to hypergraphs known results on graphs.

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“…In this section, we prove the second part of Theorem 1.6. When H is a graph, it is implied by the main result in [7]. Our result gives an alternative proof with a slight extension.…”

Section: Strong Independence Domination Numbersmentioning

confidence: 66%

Domination numbers and noncover complexes of hypergraphs

Kim

2021

Preprint

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“…The notion of d-collapsibility of simplicial complexes was introduced in [40]. In combinatorial topology it is an important problem to determine the collapsibility number or bounds for the collapsibility number of a simplicial complex and it has been widely studied (see [12,19,31,34,35]). A simple consequence of d-collapsibility is the following: Proposition 1.2.…”

Section: Introductionmentioning

confidence: 99%

On Vietoris--Rips complexes (with scale 3) of hypercube graphs

Shukla¹

2022

Preprint

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For a metric space (X, d) and a scale parameter r ≥ 0, the Vitoris-Rips complex VR(X; r) is a simplicial complex on vertex set X, where a finite set σ ⊆ X is a simplex if and only if diameter of σ is at most r. For n ≥ 1, let In denotes the n-dimensional hypercube graph. In this paper, we show that VR(In; 3) has non trivial reduced homology only in dimensions 4 and 7. Therefore, we answer a question posed by Adamaszek and Adams recently.A (finite) simplicial complex ∆ is d-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most d that is contained in a unique maximal face of ∆. The collapsibility number of ∆ is the minimum integer d such that ∆ is d-collapsible. We show that the collapsibility number of VR(In; r) is 2 r for r ∈ {2, 3}.

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On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs

Shukla

2023

SIAM J. Discrete Math.

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Collapsibility of Non-Cover Complexes of Graphs

Cited by 3 publications

References 14 publications

Domination numbers and noncover complexes of hypergraphs

Domination numbers and noncover complexes of hypergraphs

On Vietoris--Rips complexes (with scale 3) of hypercube graphs

On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs

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