Edge-only plane cubic augmentation → Plane cubic augmentation allowing new vertices Complexity open

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Clone https://github.com/reduction-zoo/plane-three-connected-cubic-augmentation to investigate Edge-only plane cubic augmentation → Plane cubic augmentation allowing new vertices. 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

Allowing new vertices changes the feasible extensions and is not automatically covered by edge-only augmentation hardness.

Known edge-only augmentation results do not automatically extend to a target that may insert vertices in the fixed embedding.

The reduction to construct

The source gives a fixed plane graph. Its outputs are three-connected cubic augmentations adding only edges while extending the embedding, or NO-SOLUTION.

The target is a fixed plane graph and asks for a three-connected cubic supergraph extending its embedding, allowing additional vertices and edges. The source version permits only additional edges. Valid augmentations require a finite combinatorial embedding encoding.

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.

Every added-vertex augmentation must decode to an edge-only source solution; prior attempts have not controlled these extra freedoms.

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 Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.