Propagation processes on (hyper)graphs: where zero forcing and burning meet

TL;DR

This paper links zero forcing and burning processes on hypergraphs, providing bounds and spectral insights, and demonstrates that the burning number is not spectrally encoded by constructing cospectral graphs with different burning numbers.

Contribution

It establishes a connection between zero forcing and burning processes via hypergraphs and explores the spectral properties related to the burning number.

Findings

Sharp upper bound on zero forcing number in terms of hypergraph burning number

Construction of cospectral graphs with different burning numbers

Insight into spectral characterization of the burning number

Abstract

The burning and forcing processes are both instances of propagation processes on graphs that are commonly used to model real-world spreading phenomena. The contribution of this paper is two-fold. We first establish a connection between these two propagation processes via hypergraphs. We do so by showing a sharp upper bound on the zero forcing number of the incidence graph of a hypergraph in terms of the lazy burning number of the hypergraph, which builds up on and improves a result by Bonato, Jones, Marbach, Mishura and Zhang (Theor. Comput. Sci., 2025). Secondly, we deepen the understanding of the role of the burning process in the context of graph spectral characterizations, whose goal is to understand which graph properties are encoded in the spectrum. While for several graph properties, including the zero forcing number, it is known that the spectrum does not encode them, this question remained open for the burning number. We solve this problem by constructing infinitely many pairs of cospectral graphs which have a different burning number.