Zero Forcing Sets in Temporal Graphs

2026-09-24

AI Summary

This study addresses the minimum zero forcing set problem in temporal graphs with dynamically evolving topologies, aiming to identify the smallest initial node set capable of infecting the entire graph. Focusing on synchronous infection mechanisms at each snapshot step, the analysis integrates graph theory, combinatorial optimization, and complexity theory. The core contributions include a rigorous proof of NP-hardness in non-strict scenarios, the development of polynomial-time exact algorithms, and a complete resolution of the open problem concerning single-step infection. By establishing a comprehensive theoretical framework that delineates computational complexity boundaries alongside efficient solution strategies, this work provides a systematic panorama of results for propagation control over dynamic graphs.

Abstract

The Zero Forcing (or corruption) of a graph is the problem of finding a minimum-size ``corrupting'' set. It corresponds to a subset of its vertices that can corrupt the whole graph by iterating the following rule: if a corrupted vertex has exactly one neighbor that is not yet corrupted, the neighbor gets corrupted. The iteration of this process comes from the fact that the corruption of a vertex might enable new corruptions (from itself or some of its neighbors). For this reason, one can consider a step of corruption, where all the possible instances of the corruption rule are applied at once. This paper investigates Zero Forcing on temporal graphs, where the topology of the graph evolves throughout the experiment. At each time step (or snapshot) of the graph, a step of corruption is resolved wherever possible. We study the problem of finding a minimum-size corrupting set such that the whole (temporal) graph is corrupted at the end of the experiment. We present a panorama of results, including NP-hardness in some not-so-restrictive scenarios, polynomial algorithms, and a solution to an open question when the whole graph must be corrupted in a single step.

Problem

• Zero Forcing
• Temporal Graphs
• Minimum Corrupting Set
• Graph Corruption

Innovation

• Zero Forcing
• Temporal Graphs
• Computational Complexity
• Polynomial Algorithms
• Graph Corruption