Characterization of the degree sequences of (quasi) regular uniform hypergraphs

TL;DR

This paper provides polynomial-time characterizations for degree sequences of (quasi) regular uniform hypergraphs, using combinatorial structures like Lyndon words and necklaces, addressing open problems in hypergraph theory.

Contribution

It introduces simple characterizations and efficient algorithms for reconstructing hypergraphs from homogeneous and almost homogeneous degree sequences.

Findings

Polynomial-time algorithms for hypergraph degree sequence characterization.

Use of Lyndon words and necklaces for hypergraph reconstruction.

Addresses open problems in hypergraph degree sequence characterization.

Abstract

In hypergraph theory, determining a characterization of the degree sequence $d=(d_1,d_2,\ldots,d_n)$ where $d_1\ge d_2\ge\ldots,d_n$ are positive integers, of an $h$-uniform simple hypergraph $\cal H$, and deciding the complexity status of the reconstruction of $\cal H$ from $d$, are two challenging open problems. They can be formulated in the context of discrete tomography: asks whether there is a matrix $A$ with positive projection vectors $H=(h,h,\ldots,h)$ and $V=(d_1,d_2,\ldots,d_n)$ with distinct rows. In this paper we consider the two subcases where the vector $V$ is an homogeneous vector, and where $V$ is almost homogeneous, i.e., $d_1-d_n=1$. We give a simple characterization for these two subcases, and we show how to solve the related reconstruction problems in polynomial time. To reach our goal, we use the concepts of Lyndon words and necklaces of fixed density, and we apply some already known algorithms for their efficient generation.