# [PFR19] On the degree sequence of 3-uniform hypergraph

**Conférence Internationale avec comité de lecture : **
21 st International Conference on Discrete Geometry for Computer Imagery,
February 2019,

pp.to appear,

*Series LNCS*,
France,

**Mots clés: ** h-uniform hypergraph, hypergraph degree sequence, discrete tomography, reconstruction problem

**Résumé: **
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 π = (d1,d2,...,dn) a non increasing sequence of positive integers, is π the degree sequence of a h-uniform simple hypergraph?
Now, assuming P ̸= N P , 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 re- search area involving discrete mathematics, discrete geometry and combinatorics.