On the Reconstruction of 3-Uniform Hypergraphs from Degree Sequences of Span-Two

Abstract

AbstractA nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence $$\pi $$ π is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for $$\pi $$ π to be k-graphic, with $$k\ge 3$$ k ≥ 3 , appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form $$\pi =(d,\ldots ,d,d-1,\ldots ,d-1,d-2,\ldots ,d-2)$$ π = ( d , … , d , d - 1 , … , d - 1 , d - 2 , … , d - 2 ) , $$d\ge 2$$ d ≥ 2 , which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to $$k \ge 4$$ k ≥ 4 and to other families of degree sequences having simple characterization, such as gap-free sequences.