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

TL;DR

This paper introduces a generalized geometric graph modification model allowing disks to be scaled within an interval, analyzing its computational complexity for various graph classes and answering open questions from prior work.

Contribution

It extends disk scaling operations to interval-based radii, providing complexity results for different graph classes and generalizing previous models.

Findings

$ ext{Pi}$-Scaling is in XP for all polynomial-time recognizable classes.

NP-hard and FPT for cluster graphs.

Polynomial-time solvable for complete graphs.

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.