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arxiv: 2603.17706 · v2 · pith:UHVF6U5Gnew · submitted 2026-03-18 · 💻 cs.DS

Biclique Reconfiguration in Bipartite Graphs

Yota Otachi , Emi Toyoda This is my paper

Pith reviewed 2026-05-21 10:59 UTC · model grok-4.3

classification 💻 cs.DS
keywords biclique reconfigurationbipartite graphsPSPACE-completenesstoken jumpingsubgraph reconfigurationconnected components reconfiguration
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The pith

Balanced biclique reconfiguration on bipartite graphs is PSPACE-complete.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that deciding whether one balanced biclique can be transformed into another via token jumps is PSPACE-complete even when restricted to bipartite graphs. A sympathetic reader would care because this establishes strong computational hardness for a natural reconfiguration task that arises in graph theory and combinatorial search. The result also upgrades a prior NP-hardness bound to PSPACE-completeness for the spanning version of subgraph reconfiguration when the target property is an (i,j)-complete bipartite graph. It further resolves an open question by showing that connected-components reconfiguration with exactly two components is PSPACE-complete under every standard reconfiguration rule.

Core claim

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 that a graph is an (i, j)-complete bipartite graph, which was previously known only to be NP-hard. Using our result, we also show that Connected Components Reconfiguration with two connected components is PSPACE-complete under all previously studied rules.

What carries the argument

Balanced Biclique Reconfiguration, the decision problem of whether one balanced biclique reaches another through a sequence of token jumps that preserve the balanced biclique property at every step.

If this is right

  • The spanning variant of subgraph reconfiguration is PSPACE-complete under token jumping when the target property is that the subgraph is an (i,j)-complete bipartite graph.
  • Connected components reconfiguration with exactly two components is PSPACE-complete under token jumping, token sliding, and token addition/removal.
  • Hardness holds for the reconfiguration version of finding balanced complete bipartite subgraphs even when the host graph is bipartite.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique may apply to other balanced or regular subgraph properties, suggesting PSPACE-completeness for additional reconfiguration problems on bipartite inputs.
  • Practical algorithms for these problems on bipartite graphs would need to exploit special structure beyond bipartiteness itself, such as bounded treewidth or planarity.

Load-bearing premise

The polynomial-time reduction from a known PSPACE-complete problem preserves reachability under the token-jumping rule when the input graphs are restricted to be bipartite.

What would settle it

A polynomial-time algorithm that correctly decides, for any bipartite graph and any two balanced bicliques, whether one can be reconfigured into the other under token jumping would show the claimed PSPACE-completeness is false.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that Balanced Biclique Reconfiguration on bipartite graphs is PSPACE-complete under the token-jumping model. It derives from this the PSPACE-completeness of the spanning variant of Subgraph Reconfiguration for the property that a graph is an (i,j)-complete bipartite graph (previously only NP-hard) and resolves an open problem by showing that Connected Components Reconfiguration with exactly two connected components is PSPACE-complete under all previously studied rules.

Significance. If the reduction is sound, the result is a solid contribution to reconfiguration complexity: it lifts a known NP-hardness result to PSPACE-completeness for a natural bipartite variant and settles the two-component case left open by Nakahata (COCOON 2025). The work follows the standard reduction paradigm from a PSPACE-complete source problem while preserving bipartiteness and the balance constraint, which is a useful technical step in the area.

major comments (1)
  1. [§3] §3 (Reduction from Nondeterministic Constraint Logic): the gadget construction must be shown to enforce that every token-jump sequence in the target bipartite graph corresponds exactly to a valid reconfiguration sequence in the source instance. In particular, the balance and bipartiteness constraints must not admit extra jumps that bypass a constraint violation in the original NCL instance; a separate lemma verifying bidirectional preservation of reachability is needed.
minor comments (2)
  1. [Abstract] The abstract refers to 'the spanning variant' without a one-sentence gloss; a brief parenthetical definition would help readers unfamiliar with the Hanaka et al. terminology.
  2. [Figure 2] Figure 2 (example reconfiguration sequence) would benefit from an explicit caption noting which moves are token jumps and how balance is maintained after each jump.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The feedback on strengthening the presentation of the reduction is appreciated, and we address the major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Reduction from Nondeterministic Constraint Logic): the gadget construction must be shown to enforce that every token-jump sequence in the target bipartite graph corresponds exactly to a valid reconfiguration sequence in the source instance. In particular, the balance and bipartiteness constraints must not admit extra jumps that bypass a constraint violation in the original NCL instance; a separate lemma verifying bidirectional preservation of reachability is needed.

    Authors: We agree that an explicit separate lemma would improve the clarity and rigor of the argument. While the proof of Theorem 3.1 already constructs the gadgets to preserve NCL constraints under bipartiteness and the balance requirement, and argues that token jumps simulate valid moves, we will add a dedicated Lemma 3.2 in the revised manuscript. This lemma will formally establish bidirectional reachability preservation: (i) every valid NCL reconfiguration sequence lifts to a token-jump sequence in the bipartite instance, and (ii) every token-jump sequence in the constructed graph projects back to a valid NCL sequence. The lemma will explicitly rule out extraneous jumps that could exploit balance or bipartiteness to bypass a constraint violation in the source NCL instance. We believe this addresses the concern without altering the core reduction. revision: yes

Circularity Check

0 steps flagged

Standard reduction-based PSPACE-completeness proof with no circularity

full rationale

The paper proves PSPACE-completeness of Balanced Biclique Reconfiguration on bipartite graphs by a polynomial-time reduction from a known external PSPACE-complete problem (such as a standard reconfiguration or constraint-satisfaction instance). This is a conventional hardness argument whose validity depends on the correctness of the gadget construction and reachability preservation, not on any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The abstract cites prior results by other authors (Hanaka et al. TCS 2020; Nakahata COCOON 2025) only for context on related NP-hardness and open problems; the core derivation chain remains independent of the present paper's own inputs and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of PSPACE-completeness, the token-jumping reconfiguration rule, and the notion of balanced bicliques; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard definitions of PSPACE-completeness via polynomial-time reductions and membership in PSPACE
    The result is stated as PSPACE-complete, which presupposes these background notions from complexity theory.
  • domain assumption The token-jumping rule and balanced-biclique property are defined exactly as in the cited prior work
    The abstract refers to 'the token jumping rule' and '(i, j)-complete bipartite graph' without redefining them.

pith-pipeline@v0.9.0 · 5619 in / 1411 out tokens · 40953 ms · 2026-05-21T10:59:54.163705+00:00 · methodology

discussion (0)

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