Reconstruction of hypergraphs from line graphs and degree sequences

TL;DR

This paper investigates the problem of reconstructing hypergraphs from line graphs and degree sequences, providing polynomial-time solutions under certain conditions and highlighting NP-completeness in general.

Contribution

It offers a polynomial-time method for reconstructing hypergraphs with constant degree sequences and addresses the complexity of the problem when hyperedges are bounded.

Findings

Reconstruction is feasible in polynomial time for constant degree sequences.

The problem is NP-complete for arbitrary degree sequences.

Baranyai's theorem is utilized to achieve polynomial-time solutions.

Abstract

In this paper we consider the problem to reconstruct a $k$-uniform hypergraph from its line graph. In general this problem is hard. We solve this problem when the number of hyperedges containing any pair of vertices is bounded. Given an integer sequence, constructing a $k$-uniform hypergraph with that as its degree sequence is NP-complete. Here we show that for constant integer sequences the question can be answered in polynomial time using Baranyai's theorem.