Structure and Complexity of 2-Intersection Graphs of 3-Hypergraphs

Given a 3-uniform hypergraph H having a set V of vertices, and a set of hyperedges T subset of P(V), whose elements have cardinality three each, a null labelling is an assignment of +/- 1 to the hyperedges such that each vertex belongs to the same number of hyperedges labelled +1 and -1. A sufficient condition for the existence of a null labelling of H (proved in Di Marco et al. Lect Notes Comput Sci 12757:282-294, 2021) is a Hamiltonian cycle in its 2-intersection graph. The notion of 2-intersection graph generalizes that of intersection graph of an (hyper)graph and extends its effectiveness. The present study first shows that this sufficient condition for the existence of a null labelling in H can not be weakened by requiring only the connectedness of the 2-intersection graph. Then some interesting properties related to their clique configurations are proved. Finally, the main result is proved, the NP-completeness of this characterization and, as a consequence, of the construction of the related 3-hypergraphs.