On null 3-hypergraphs

A 3-uniform hypergraph H consists of a set V of vertices, and E ⊆ V 3 triples. Let a null labelling be an assignment of ±1 to the triples such that each vertex has signed degree equal to zero. Assumed as necessary condition the degree of every vertex of H to be even, the Null Labelling Problem consists in determining whether H has a null labelling. Although the problem is NP-complete, the subclasses where the problem turns out to be polynomially solvable are of interest. In this study we define the notion of 2-intersection graph related to a 3-uniform hypergraph, and we prove that the existence of a Hamiltonian cycle there, is sufficient to obtain a null labelling in the related hypergraph. The proof we propose provides an efficient way of computing the null labelling.

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“…This is an extension of the idea of a line graph to 3-hypergraphs. In [8] the following result was proved.…”

Section: Lemmamentioning

confidence: 89%

“…This raises the question of whether there is a characterization of null hypergraphs. In [8] it is shown that the problem of finding a null labelling even for the simplest case of 3-hypergraphs is NP-complete.…”

Section: Lemmamentioning

confidence: 99%

A Study on the Existence of Null Labelling for 3-Hypergraphs

Marco

Frosini

Kocay

2021

Lecture Notes in Computer Science

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“…These switch operations are clearly analogous to the switch operations of simple and bipartite graphs. We also remark that these switch operations are the tripartite hypergraph versions of the N 6 null-hypergraphs introduced by Kocay and Li [ 25 ], see also [ 26 ].…”

Section: Realizing Hypergraphic Degree Sequencesmentioning

confidence: 99%

Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence

Hubai,

Mezei,

Béres

et al. 2024

PLoS ONE

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Partite, 3-uniform hypergraphs are 3-uniform hypergraphs in which each hyperedge contains exactly one point from each of the 3 disjoint vertex classes. We consider the degree sequence problem of partite, 3-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is k or k − 1 for some fixed k, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, 3-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for χ2 testing of contingency tables. We have shown that this hypergraph-based χ2 test is more sensitive than the standard χ2 test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.

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“…This raises the question of whether there is a characterization of null (hyper)graphs. From a computational perspective, the null labelling problems on k-hypergraphs turns out to be NP-hard when k ≥ 3, as proved in [8].…”

Section: Introductionmentioning

confidence: 95%

“…As an example, in [12], the authors defined a general operator to move among all the 3-hypergraphs sharing the same degree sequence and they introduced the notion of null-labelling obtained as the symmetric difference of the edges of any two 3-hypergraphs. In [8], this notion, restricted to 3-hypergraphs, is studied through their intersection graph and a sufficient condition for its existence was provided. Finally, in [6], it was shown that as the number of edges of the related 3-hypergraph increases, the usefulness of the intersection graph decreases, and the stronger notion of 2-intersection graph was introduced.…”

Section: Introductionmentioning

confidence: 99%

Structure and Complexity of 2-Intersection Graphs of 3-Hypergraphs

et al. 2022

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Given a 3-uniform hypergraph H having a set V of vertices, and a set of hyperedges $$T\subset \mathcal {P}(V)$$ T ⊂ P ( V ) , whose elements have cardinality three each, a null labelling is an assignment of $$\pm 1$$ ± 1 to the hyperedges such that each vertex belongs to the same number of hyperedges labelled $$+1$$ + 1 and $$-1$$ - 1 . A sufficient condition for the existence of a null labelling of H (proved in Di Marco et al. Lect Notes Comput Sci 12757:282–294, 2021) is a Hamiltonian cycle in its 2-intersection graph. The notion of 2-intersection graph generalizes that of intersection graph of an (hyper)graph and extends its effectiveness. The present study first shows that this sufficient condition for the existence of a null labelling in H can not be weakened by requiring only the connectedness of the 2-intersection graph. Then some interesting properties related to their clique configurations are proved. Finally, the main result is proved, the NP-completeness of this characterization and, as a consequence, of the construction of the related 3-hypergraphs.

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On null 3-hypergraphs

Cited by 5 publications

References 8 publications

A Study on the Existence of Null Labelling for 3-Hypergraphs

A Study on the Existence of Null Labelling for 3-Hypergraphs

Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence

Structure and Complexity of 2-Intersection Graphs of 3-Hypergraphs

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