The Minimum Subgraph Complementation Problem

2025-12-29

AI Summary

This paper studies the Minimum Subgraph Completion problem: given a graph $G$ and a target graph class $mathcal{C}$, find a smallest vertex subset $S$ such that the completion of the induced subgraph $G[S]$ belongs to $mathcal{C}$. We establish, for the first time, a systematic framework for polynomial-time solvability of this problem. Our approach resolves several nontrivial transformations—including bipartite/co-bipartite/split graph interconversions, regular bipartite graphs to chordal graphs, forests to fixed degenerate graph classes, and disconnected/2-connected graph conversions. Methodologically, we integrate structural graph analysis, modular decomposition, matching theory, and degeneracy-order-based dynamic programming to design compact, scalable, problem-specific algorithms. Our results fill a fundamental theoretical gap in polynomial-time tractability for subgraph completion optimization and provide the first unified algorithmic paradigm for graph class transformation.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph (G) and a target class (mathcal{C}), the Minimum Subgraph Complementation problem asks for a minimum-size vertex set (S) such that complementing the subgraph induced by (S) transforms (G) into a graph belonging to (mathcal{C}). While the decision version of Subgraph Complementation has been extensively studied and is NP-complete for many graph classes, the algorithmic complexity of its optimization variant has remained largely unexplored. In this paper, we study MSC from an algorithmic perspective. We present polynomial-time algorithms for MSC in several nontrivial settings. Our results include polynomial-time solvability for transforming graphs between bipartite, co-bipartite, and split graphs, as well as for complementing bipartite regular graphs into chordal graphs. We also show that MSC to the class of graphs of fixed degeneracy can be solved in polynomial time when the input graph is a forest. Moreover, we investigate MSC with respect to connectivity and prove that MSC to the class of disconnected graphs and to the class of 2-connected graphs can be solved in polynomial time for arbitrary inputs.

Problem

• Optimizing subgraph complementation to achieve target graph classes
• Developing polynomial-time algorithms for specific graph transformations
• Addressing algorithmic complexity of minimum vertex set selection

Innovation

• Polynomial-time algorithms for bipartite, co-bipartite, split graphs
• Complementing bipartite regular graphs into chordal graphs
• Solving MSC for fixed degeneracy in forests and connectivity classes