Degree-four Matching Cut → Line-graph 2-Cut Complexity open

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

Solution submitted Research history & proof on GitHub ↗

Why this question matters

The d=2 case tests a missing boundary for degree-bounded cuts on line graphs.

The cited d-Cut classification retains the line-graph case d=2; results for other values of d do not settle this endpoint.

The reduction to construct

The source is a connected simple graph of maximum degree at most four. Its outputs are nontrivial vertex bipartitions with at most one opposite neighbor per vertex, or NO-SOLUTION.

The source is a connected simple graph of maximum degree four and seeks a nontrivial cut of crossing degree at most one. The target is a connected line graph and seeks a nontrivial cut of crossing degree at most two.

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 root-graph interface must preserve every valid cut, including cuts not aligned with intended source vertices.

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 Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.