What was already known
Bentz posed the optimization question in 2009. Polynomial-time algorithms were known for ordinary vertex outerplanarity and minimum face-depth.
AI involvement
GPT 5.5 Pro produced the initial proof, Claude Opus 4.8 reviewed it, and Hantao Yu manually verified and polished the argument.
The source directly credits GPT 5.5 Pro as solver of the initial complete proof, followed by model and author verification.
Importance
The reduction answers Bentz’s algorithmic question for loopless planar graphs.
Limitations
The paper gives a polynomial-time guarantee but does not state a tighter runtime for the final reduction or detail every model-to-human proof edit.
Technical verification
The author expressly says he verified the generated proof before publication. This is a source-reported completed human check of Theorem 1.1; the Claude verifier is not counted as a machine proof check.
Formal checking
Manual verification and polishing of the GPT-generated proof
- Checker
- Hantao Yu, author
- Scope
- Theorem 1.1 polynomial-time minimum edge-outerplanar embedding for finite loopless planar graphs
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Yu explains why edge-peeling layers equal face-adjacency depth and why the auxiliary graph preserves the optimum, including disconnected components.
Correction and publication history
The reduction answers Bentz’s algorithmic question for loopless planar graphs.
Read this published revision