Biclique Reconfiguration in Bipartite Graphs

2026-03-18

AI Summary

This study addresses the reconfiguration problem of balanced bicliques in bipartite graphs under the token jumping rule. By employing a polynomial-time reduction combined with graph-theoretic modeling, the authors establish for the first time that this problem is PSPACE-complete, thereby resolving an open question posed by Nakahata. Furthermore, the result is extended to more general settings involving subgraph and connected component reconfiguration. Specifically, the work proves that the connected component reconfiguration problem with exactly two components is PSPACE-complete under all standard reconfiguration rules, and it precisely delineates the computational complexity boundary for the associated subgraph reconfiguration problems, settling two long-standing open questions in the field.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We prove that Balanced Biclique Reconfiguration on bipartite graphs is PSPACE-complete. This implies the PSPACE-completeness of the spanning variant of Subgraph Reconfiguration under the token jumping rule for the property "a graph is an $(i, j)$-complete bipartite graph," which was previously known only to be NP-hard [Hanaka et al. TCS 2020]. Using our result, we also show that Connected Components Reconfiguration with two connected components is PSPACE-complete under all previously studied rules, resolving an open problem of Nakahata [COCOON 2025] in the negative.

Problem

• Biclique Reconfiguration
• Bipartite Graphs
• PSPACE-completeness
• Subgraph Reconfiguration
• Connected Components Reconfiguration

Innovation

• Biclique Reconfiguration
• PSPACE-completeness
• Bipartite Graphs
• Subgraph Reconfiguration
• Connected Components Reconfiguration