Bipartite radius-four Matching Cut → Bipartite radius-three Matching Cut Complexity open

Model and version: gpt-6-astra
Date: Not recorded

Solution submitted Research history & proof on GitHub ↗

Why this question matters

The question asks whether one fewer distance layer crosses the tractability boundary for Matching Cut.

The published radius classification does not settle the bipartite radius-three case using its radius-four construction.

The reduction to construct

The source is a connected bipartite graph of radius at most four. Its outputs are nontrivial cuts with crossing degree at most one, or NO-SOLUTION.

Both problems ask for a nontrivial vertex bipartition with at most one opposite neighbor per vertex, or NO-SOLUTION. Inputs are connected bipartite graphs, with radius at most four in the source and at most three in the target.

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.

A communication structure must shorten distances without creating a cut unrelated to the source.

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 2026-09-15. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.