Revisiting Graph Modification via Disk Scaling: From One Radius to Interval-Based Radii

2026-03-05

AI Summary

This work addresses the problem of transforming an input graph into a target graph class while preserving its inherent geometric structure, by introducing a more flexible radius adjustment scheme. The key innovation lies in replacing the conventional fixed radius with an interval-based radius, thereby establishing a general disk-scaling framework that significantly enhances the capacity of geometric graph modification models to handle geometric information. Through parameterized complexity analysis, polynomial-time recognition algorithms, and reduction techniques, the authors demonstrate that the Π-Scaling problem is in XP for any polynomially recognizable graph class; it is NP-hard but fixed-parameter tractable (FPT) for cliques, polynomial-time solvable for complete graphs, and W[1]-hard for connected graphs, thereby resolving several previously open questions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

For a fixed graph class $\Pi$, the goal of $\Pi$-Modification is to transform an input graph $G$ into a graph $H\in\Pi$ using at most $k$ modifications. Vertex and edge deletions are common operations, and their (parameterized) complexity for various $\Pi$ is well-studied. Classic graph modification operations such as edge deletion do not consider the geometric nature of geometric graphs such as (unit) disk graphs. This led Fomin et al. [ITCS'25] to initiate the study of disk scaling as a geometric graph modification operation for unit disk graphs: For a given radius $r$, each modified disk will be rescaled to radius $r$. In this paper, we generalize their model by allowing rescaled disks to choose a radius within a given interval $[r_{\min}, r_{\max}]$ and study the (parameterized) complexity (with respect to $k$) of the corresponding problem $\Pi$-Scaling. We show that $\Pi$-Scaling is in XP for every graph class $\Pi$ that can be recognized in polynomial time. Furthermore, we show that $\Pi$-Scaling: (1) is NP-hard and FPT for cluster graphs, (2) can be solved in polynomial time for complete graphs, and (3) is W[1]-hard for connected graphs. In particular, (1) and (2) answer open questions of Fomin et al. and (3) generalizes the hardness result for their variant where the set of scalable disks is restricted.

Problem

• graph modification
• disk scaling
• unit disk graphs
• interval-based radii
• parameterized complexity

Innovation

• disk scaling
• interval-based radii
• parameterized complexity
• graph modification
• geometric graphs