For a fixed graph class $Π$, the goal of $Π$-Modification is to transform an input graph $G$ into a graph $H\inΠ$ using at most $k$ modifications. Vertex and edge deletions are common operations, and their (parameterized) complexity for various $Π$ 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 $Π$-Scaling. We show that $Π$-Scaling is in XP for every graph class $Π$ that can be recognized in polynomial time. Furthermore, we show that $Π$-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.

翻译：对于一个固定的图类 $Π$，$Π$-修改问题的目标是通过最多 $k$ 次修改将输入图 $G$ 变换为一个图 $H\inΠ$。顶点和边删除是常见的操作，并且针对不同 $Π$ 的（参数化）复杂性已得到充分研究。经典的图修改操作（如边删除）并未考虑几何图（如（单位）圆盘图）的几何性质。这促使 Fomin 等人 [ITCS' 25] 开创性地将圆盘缩放作为单位圆盘图的一种几何图修改操作进行研究：对于给定的半径 $r$，每个被修改的圆盘将被重新缩放到半径 $r$。在本文中，我们推广了他们的模型，允许缩放后的圆盘在给定区间 $[r_{\min}, r_{\max}]$ 内选择一个半径，并研究了相应问题 $Π$-缩放的（参数化）复杂性（关于参数 $k$）。我们证明，对于每个可在多项式时间内识别的图类 $Π$，$Π$-缩放均属于 XP 类。此外，我们证明 $Π$-缩放：（1）对于聚类图是 NP 难且 FPT 的，（2）对于完全图可在多项式时间内求解，以及（3）对于连通图是 W[1]-难的。特别地，（1）和（2）回答了 Fomin 等人提出的开放性问题，而（3）推广了其变体（其中可缩放圆盘集合受限）的难度结果。