Burning Signed Graphs

2026-10-01

AI Summary

This study addresses the problem of competitive two-fire propagation on signed graphs to maximize the number of monochromatic burned vertices. It introduces a novel two-fire competitive signed graph burning model in which edge signs govern fire color transitions. By integrating binary-encoded Hamming distance analysis with multimodality, this work establishes a theoretical connection between multimodality and coding covering radius, supported by rigorous proofs grounded in computational complexity theory. The contributions include precisely determining optimal values on path graphs and revealing intrinsic relationships between multimodality and coding theory. Furthermore, it proves that both exact optimization and approximation on general graphs are NP-hard. Ultimately, this research provides a new theoretical framework for understanding competitive propagation mechanisms in networks.

Abstract

We introduce and analyze a new model of graph burning, in which two competing fires (coloured yellow and green) ignite vertices of a given signed graph and propagate along its edges. The boolean sign of an edge determines whether a fire spreading along the edge changes colour or not. In each step, a player ignites a new vertex in a colour of their choice, while previously activated fires continue to spread. Given a signed graph $Γ$, the objective is to burn the maximum possible number of vertices in a single colour; the optimal achievable value is called the plurality number of $Γ$. We express the plurality number through Hamming distances to a binary code associated with $Γ$. In particular, the minimum plurality number over the switching class of $Γ$ equals the number of vertices minus the covering radius of this code. Under certain conditions on the signature, we determine exact values of the plurality number of signed paths. By contrast, we prove hardness results for multiple variants of the problem of determining or approximating the plurality number.

Problem

• Signed Graphs
• Graph Burning
• Plurality Number
• Computational Complexity

Innovation

• Signed Graphs
• Graph Burning
• Plurality Number
• Binary Code
• Covering Radius