On the Degree Sequence of 3-Uniform Hypergraph: A New Sufficient Condition
The study of the degree sequences of h-uniform hypergraphs, say h-sequences, was a longstanding open problem in the case of \(h>2\), until very recently where its decision version was proved to be NP-complete. Formally, the decision version of this problem is: Given \(\pi =(d_1,d_2,\ldots ,d_n)\) a non increasing sequence of positive integers, is \(\pi \) the degree sequence of a h-uniform simple hypergraph?
Now, assuming \(P\ne NP\), we know that such an effective characterization cannot exist even for the case of 3-uniform hypergraphs.
However, several necessary or sufficient conditions can be found in the literature; here, relying on a result of S. Behrens et al., we present a sufficient condition for the 3-graphicality of a degree sequence and a polynomial time algorithm that realizes one of the associated 3-uniform hypergraphs, if it exists. Both the results are obtained by borrowing some mathematical tools from discrete tomography, a quite recent research area involving discrete mathematics, discrete geometry and combinatorics.
Keywordsh-uniform hypergraph Hypergraph degree sequence Discrete tomography Reconstruction problem
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