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

TL;DR

The paper studies two propagation mechanisms, zero forcing and burning, on graphs and hypergraphs, and connects them through the incidence graph construction. It proves a sharp bound, $Z(IG(H)) \leq b_L(H) + |E(H)| - k$, where $k$ is the number of connected components of $H$ containing at least one non-singleton hyperedge, with refinements for component structure, linking lazy burning to zero forcing. In the spectral realm, the authors show that the burning number is not determined by the adjacency spectrum by constructing infinite families of cospectral graphs with differing burning numbers via strong products with complete graphs. The results bridge hypergraph burning and graph-zero forcing theory and demonstrate that burning numbers can escape spectral characterization, highlighting fundamental differences between these propagation processes.

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.

Figures (8)

• Figure 1: A hypergraph with 4 vertices and 3 hyperedges.
• Figure 2: The incidence graph of the hypergraph in Figure \ref{['fig:hypergraph-ex']}.
• Figure 3: Graph burning process for the first three rounds with burned vertices marked in red and sources are highlighted in deep red.
• Figure 4: The lazy burning process on a hypergraph.
• Figure 5: An improved zero forcing set on the incidence graph of the hypergraph in Figure \ref{['fig:hypergraph-ex']}.

Theorems & Definitions (22)

• Theorem 2.1
• Lemma 2.2
• Example 2.3
• Theorem 2.4
• Example 2.5
• Theorem 3.1
• Example 3.2
• Theorem 3.3
• proof
• Example 3.4