Properties of Unique Degree Sequences of 3-Uniform Hypergraphs
| Autor: | William L. Kocay, Michela Ascolese, Andrea Frosini, Lama Tarsissi |
|---|---|
| Přispěvatelé: | Laboratoire d'Informatique Gaspard-Monge (LIGM), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Sequence
Hypergraph Mathematics::Combinatorics 021103 operations research Conjecture Degree (graph theory) 0211 other engineering and technologies Integer sequence 0102 computer and information sciences 02 engineering and technology [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] 01 natural sciences Combinatorics 010201 computation theory & mathematics Analysis of algorithms Graphic sequence Uniqueness problem Time complexity Maximal element ComputingMilieux_MISCELLANEOUS Mathematics |
| Zdroj: | Discrete Geometry and mathematica Discrete Geometry and mathematica, pp.312-324, 2021, ⟨10.1007/978-3-030-76657-3_22⟩ Lecture Notes in Computer Science ISBN: 9783030766566 DGMM |
| DOI: | 10.1007/978-3-030-76657-3_22⟩ |
| Popis: | In 2018 Deza et al. proved the NP-completeness of deciding wether there exists a 3-uniform hypergraph compatible with a given degree sequence. A well known result of Erdos and Gallai (1960) shows that the same problem related to graphs can be solved in polynomial time. So, it becomes relevant to detect classes of uniform hypergraphs that are reconstructible in polynomial time. In particular, our study concerns 3-uniform hypergraphs that are defined in the NP-completeness proof of Deza et al. Those hypergraphs are constructed starting from a non-increasing sequence s of integers and have very interesting properties. In particular, they are unique, i.e., there do not exist two non isomorphic 3-uniform hypergraphs having the same degree sequence \(d_s\). This property makes us conjecture that the reconstruction of these hypergraphs from their degree sequences can be done in polynomial time. So, we first generalize the computation of the \(d_s\) degree sequences by Deza et al., and we show their uniqueness. We proceed by defining the equivalence classes of the integer sequences determining the same \(d_s\) and we define a (minimal) representative. Then, we find the asymptotic growth rate of the maximal element of the representatives in terms of the length of the sequence, with the aim of generating and then reconstructing them. Finally, we show an example of a unique 3-uniform hypergraph similar to those defined by Deza et al. that does not admit a generating integer sequence s. The existence of this hypergraph makes us conjecture an extended generating algorithm for the sequences of Deza et al. to include a much wider class of unique 3-uniform hypergraphs. Further studies could also include strategies for the identification and reconstruction of those new sequences and hypergraphs. |
| Databáze: | OpenAIRE |
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