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Biclique Reconfiguration in Bipartite Graphs

Yota Otachi, Emi Toyoda

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.

Biclique Reconfiguration in Bipartite Graphs

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 -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.
Paper Structure (13 sections, 10 theorems, 3 equations)

This paper contains 13 sections, 10 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Confronting the bipartiteness
  3. Our results
  4. The main result
  5. The second result: an application of the main result
  6. Preliminaries
  7. Balanced Biclique Reconfiguration on bipartite graphs
  8. The bipartite graph $H$
  9. The bicliques $S$ and $S'$
  10. Connected Components Reconfiguration with two connected components
  11. Concluding remarks
  12. Token sliding
  13. Flexible variant

Key Result

Theorem 1.1

Balanced Biclique Reconfiguration is $\mathrm{PSPACE}$-complete on bipartite graphs.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • proof
  • Lemma 3.3: Johnson87
  • proof
  • Lemma 3.4
  • proof
  • ...and 9 more