Infection in Hypergraphs

TL;DR

This paper introduces the hypergraph infection parameter, generalizing zero forcing in graphs, and provides exact values and bounds for various hypergraph classes, advancing understanding of infection dynamics in complex hyperstructures.

Contribution

It defines hypergraph infection, extends zero forcing concepts, and derives formulas and bounds for infection numbers in multiple hypergraph families.

Findings

Exact infection numbers for complete hypergraphs

Formulas for interval hypergraphs and cyclic hypergraphs

Bounds for hypergraphs with t-design edges

Abstract

In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is determined. A formula for the infection number for interval hypergraphs and several families of cyclic hypergraphs is given. The value of the infection number for a hypergraph whose edges form a symmetric t-design is given, and bounds are determined for a hypergraph whose edges are a t-design. Finally, the infection number for several hypergraph products and line graphs are considered.