Extending Graph Burning to Hypergraphs

TL;DR

This paper generalizes the graph burning process to hypergraphs, introduces a lazy variant, and explores their properties, bounds, and complexities, revealing differences from traditional graph burning and posing open problems.

Contribution

It extends graph burning to hypergraphs, analyzes lazy burning complexity, and establishes bounds and open problems specific to hypergraph structures.

Findings

Lazy hypergraph burning can be complex, unlike trivial lazy graph burning.

Hypergraphs do not satisfy the same bounds as graphs in the Burning Number Conjecture.

Bounds on hypergraph burning numbers are derived in terms of hypergraph parameters.

Abstract

Graph burning is a round-based game or process that discretely models the spread of influence throughout a network. We introduce a generalization of graph burning which applies to hypergraphs, as well as a variant called ''lazy'' hypergraph burning. Interestingly, lazily burning a graph is trivial, while lazily burning a hypergraph can be quite complicated. Moreover, the lazy burning model is a useful tool for analyzing the round-based model. One of our key results is that arbitrary hypergraphs do not satisfy a bound analogous to the one in the Burning Number Conjecture for graphs. We also obtain bounds on the burning number and lazy burning number of a hypergraph in terms of its parameters, and present several open problems in the field of (lazy) hypergraph burning.