3P2-free 2-Cut → 2P4-free 2-Cut Complexity open

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Clone https://github.com/reduction-zoo/two-p4-free-two-cut to investigate 3P2-free 2-Cut → 2P4-free 2-Cut. Read the fixed question and state. Use preparation notes, the test corpus, and verifier if present; otherwise complete Prepare before a reduction attempt. Follow the repository instructions and your AutoResearch pipeline.

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Why this question matters

This addresses a remaining forbidden-induced-subgraph case for bounded-degree cuts.

The cited induced-subgraph classification leaves the 2P4-free case distinct from the known 3P2-free source case.

The reduction to construct

The source is a connected graph containing no induced disjoint union of three edges. Its outputs are nontrivial cuts with crossing degree at most two, or NO-SOLUTION.

Both problems seek a nontrivial cut with at most two opposite neighbors per vertex in a connected simple graph. The source is 3P2-free and the target has no induced disjoint union of two four-vertex paths.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is NP-hardness for the stated target problem.

Connecting gadgets must eliminate pairs of induced paths while preserving all legal cut choices.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-14. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.