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arxiv: 2606.31815 · v1 · pith:VDRI6TBAnew · submitted 2026-06-30 · 💻 cs.DS · math.CO

Token sliding independent set reconfiguration on graphs with few P₄'s

Lucia Busolini , Mario Valencia-Pabon This is my paper

Pith reviewed 2026-07-01 02:33 UTC · model grok-4.3

classification 💻 cs.DS math.CO
keywords independent set reconfigurationtoken slidingP4-tidy graphs(q,q-4)-graphspolynomial time algorithmcographsgraph algorithmsreconfiguration problems
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The pith

Token-sliding independent set reconfiguration admits polynomial-time algorithms on P4-tidy graphs and (q,q-4)-graphs.

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

The paper gives polynomial-time algorithms for deciding reachability between two independent sets under the token-sliding rule on P4-tidy graphs and on (q,q-4)-graphs. Both classes generalize cographs by restricting the induced P4 subgraphs that can appear. The problem is PSPACE-hard on split graphs, planar graphs, and bounded-treewidth graphs, yet the new algorithms exploit the limited-P4 structure to decide reachability efficiently. A reader cares because these classes include many graphs that arise in applications where exact reconfiguration decisions must remain feasible.

Core claim

The authors show that the token-sliding independent set reconfiguration problem can be solved in polynomial time for every P4-tidy graph and every (q,q-4)-graph by using the structural decompositions available in these families to search for valid sequences of slides.

What carries the argument

Structural decompositions of P4-tidy graphs and (q,q-4)-graphs that support recursive or dynamic-programming search over independent-set configurations.

If this is right

  • Reachability under token slides is decidable in polynomial time for every graph in these two classes.
  • The same structural properties that make cographs tractable extend to these strictly larger families.
  • Any graph class properly containing P4-tidy graphs or (q,q-4)-graphs may still contain hard instances.
  • The algorithms provide explicit certificates of reachability or non-reachability when they terminate.

Where Pith is reading between the lines

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

  • The presence of many induced P4s may be the feature that drives PSPACE-hardness in broader graph classes.
  • Similar decomposition techniques could be tested on other token-moving reconfiguration problems such as vertex cover or dominating set on the same graph families.
  • One could measure the practical running time of the algorithms on randomly generated graphs whose P4 density is controlled to lie just inside or outside the two classes.

Load-bearing premise

The input graphs belong to the P4-tidy or (q,q-4) classes and therefore possess the decompositions the algorithm requires.

What would settle it

A concrete P4-tidy graph together with two independent sets I and I' on which the algorithm reports no reconfiguration sequence exists, yet an explicit sequence of token slides transforming I into I' can be exhibited.

Figures

Figures reproduced from arXiv: 2606.31815 by Lucia Busolini, Mario Valencia-Pabon.

Figure 1
Figure 1. Figure 1: Possible quasi-starfishes of size two. From left to right: [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We consider the INDEPENDENT SET RECONFIGURATION problem under the Token Sliding rule. Let $I$ be an independent set of a simple undirected graph $G$. Suppose that each vertex of $I$ has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of $G$, so that after each move, the vertices having tokens always form an independent set of $G$. The problem we deal is to decide if we can transform $I$ into $I'$ through a sequence of steps, each of which involves substituting a vertex in the current independent set with one of its neighbours to obtain another independent set. This problem of determining if one independent set of a graph "is reachable" from another independent set of it is known to be PSPACE-hard even for split graphs, planar graphs, and graphs of bounded treewidth. Polynomial time algorithms have been obtained for certain graph classes like trees, interval graphs, claw-free graphs, bipartite permutation graphs, block graphs, and cographs. We present a polynomial time algorithm for the problem on $P_4$-tidy graphs and $(q,q-4)$-graphs, both families of graphs generalizing cographs.

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 / 0 minor

Summary. The manuscript asserts the existence of a polynomial-time algorithm for the token-sliding independent set reconfiguration problem on P4-tidy graphs and (q,q-4)-graphs, both of which generalize cographs and admit cotree-like decompositions.

Significance. If the algorithm is correct, the result would modestly extend the known tractable cases for this PSPACE-hard problem by leveraging the bounded-P4 structural decompositions already established for these classes; however, the significance cannot be fully evaluated without a correctness argument or running-time analysis.

major comments (1)
  1. Abstract: the claim that a polynomial-time algorithm exists is made without any proof sketch, running-time analysis, or correctness argument, so the soundness of the central claim cannot be assessed from the provided information.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

  1. Referee: [—] Abstract: the claim that a polynomial-time algorithm exists is made without any proof sketch, running-time analysis, or correctness argument, so the soundness of the central claim cannot be assessed from the provided information.

    Authors: Abstracts are concise summaries and are not intended to contain full proofs, sketches, or analyses; those appear in the body of the manuscript. The full paper provides a detailed polynomial-time algorithm for token-sliding independent set reconfiguration on P4-tidy graphs and (q,q-4)-graphs. It leverages the known cotree-like decompositions for these classes, describes the algorithm explicitly, includes a correctness proof, and analyzes the running time. revision: no

Circularity Check

0 steps flagged

No significant circularity in algorithmic claim

full rationale

The paper presents a polynomial-time algorithm for token-sliding independent set reconfiguration on P4-tidy graphs and (q,q-4)-graphs, extending known results for cographs via their structural decompositions. No equations, fitted parameters, predictions, or self-referential derivations appear; the result is an algorithmic existence claim relying on graph class properties rather than reducing to its own inputs by construction. No load-bearing self-citations or ansatzes are invoked in the provided abstract and context.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are identifiable from the abstract; the result rests on the standard definitions of the two graph classes and the token-sliding rule.

pith-pipeline@v0.9.1-grok · 5750 in / 1081 out tokens · 46332 ms · 2026-07-01T02:33:17.038723+00:00 · methodology

discussion (0)

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Reference graph

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