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

TL;DR

This paper generalizes the study of disk scaling as a geometric graph modification operation for unit disk graphs by allowing rescaled disks to choose a radius within a given interval and shows that $\Pi$-Scaling is in XP for every graph class $\Pi$ that can be recognized in polynomial time.

Abstract

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.

Figures (11)

• Figure 1: (a) Turning the underlying abstract graph into a $K_{1,6}$ by adding a single edge is easy. However, this is not possible while maintaining a unit disk graph as (b) demonstrates.
• Figure 2: We can turn (a) into a cluster graph by scaling the blue disk as shown in (b). However, if we enlarge the disk too much as in (c), the resulting disk graph might no longer be a cluster graph.
• Figure 3: The (parameterized) complexity of $\Pi$-Scaling. Bold boxes are results from this paper and the remaining follow from FG00Z25. †: Assuming that recognizing graphs in $\Pi$ takes polynomial time.
• Figure 4: Let $\mathcal{T} = \{p^*,q\}$. The furthest unscaled neighbor $\mathsf{far}(p^*)$ (purple) of $p^*$ (blue) determines if $p^*p' \in E(H)$ for $p' \in \mathcal{S} \setminus \mathcal{T}$. For the edge to $q$ (red), we branch to fix if it exists (as here) or not. The disk graph (a) before and (b) after the scaling operations.
• Figure 5: Overview of our -algorithm for Scaling To Cluster with a small example. (a) An input instance and the corresponding edge-colored complete graph $H_0$ (we omit red edges for clarity). (b)--(d) In Phase 1, we iteratively identify disks to scale (the sets $\mathcal{T}_i$) and refine our edge-colored complete graph $H_i$ (changes highlighted in bold). (e) In Phase 2, we branch to fix a compatible cluster graph $H$, and check using \ref{['prop:radius-feasibility-running-time']} if it can be realized. (f) The solution to (a).

Theorems & Definitions (21)

• Lemma 1
• Proposition 2
• Theorem 3
• Remark 4
• Lemma 5
• Theorem 9
• Lemma 14
• Proposition 15
• Proposition 18
• Remark 19