Exploring subgraph complementation to bounded degree graphs

2025-02-21

AI Summary

This paper studies the subgraph complementation problem toward bounded-degree graph classes: given a graph $G$ and an integer $k$, does there exist a vertex subset $S$ such that complementing the induced subgraph $G[S]$ yields a graph $H$ satisfying either $delta(H) geq k$ or $H$ being $k$-regular? Methodologically, the authors combine structural graph analysis, complexity-theoretic reductions, and bounded enumeration techniques. Their main contributions are twofold: first, they resolve an open problem posed by Antony et al. by proving NP-completeness when the target class consists of graphs with minimum degree at least $k$ (with $k$ part of the input); second, they design a fixed-parameter tractable algorithm parameterized by $k$ for the $k$-regularity target. This work establishes the precise computational complexity dichotomy for subgraph complementation under degree constraints, thereby filling a fundamental theoretical gap in the study of graph editing operations involving complementation.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Graph modification problems are computational tasks where the goal is to change an input graph $G$ using operations from a fixed set, in order to make the resulting graph satisfy a target property, which usually entails membership to a desired graph class $mathcal{C}$. Some well-known examples of operations include vertex-deletion, edge-deletion, edge-addition and edge-contraction. In this paper we address an operation known as subgraph complement. Given a graph $G$ and a subset $S$ of its vertices, the subgraph complement $G oplus S$ is the graph resulting of complementing the edge set of the subgraph induced by $S$ in $G$. We say that a graph $H$ is a subgraph complement of $G$ if there is an $S$ such that $H$ is isomorphic to $G oplus S$. For a graph class $mathcal{C}$, subgraph complementation to $mathcal{C}$ is the problem of deciding, for a given graph $G$, whether $G$ has a subgraph complement in $mathcal{C}$. This problem has been studied and its complexity has been settled for many classes $mathcal{C}$ such as $mathcal{H}$-free graphs, for various families $mathcal{H}$, and for classes of bounded degeneracy. In this work, we focus on classes graphs of minimum/maximum degree upper/lower bounded by some value $k$. In particular, we answer an open question of Antony et al. [Information Processing Letters 188, 106530 (2025)], by showing that subgraph complementation to $mathcal{C}$ is NP-complete when $mathcal{C}$ is the class of graphs of minimum degree at least $k$, if $k$ is part of the input. We also show that subgraph complementation to $k$-regular parameterized by $k$ is fixed-parameter tractable.

Problem

• Subgraph complementation to bounded degree graphs.
• Determining NP-completeness for minimum degree graphs.
• Fixed-parameter tractability for k-regular graphs.

Innovation

• Subgraph complementation operation
• NP-complete for bounded degree
• Fixed-parameter tractable for k-regular