Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Maximal plane graphs, plane triangulations, and maximally planar abstract graphs
Definition
A plane graph (Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs) is maximal plane if no edge can be added between two nonadjacent existing vertices while preserving a plane embedding with the same vertex positions. It is a plane triangulation if every face, including the unbounded face, has boundary a triangle in the sense of Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices.
An abstract simple planar graph is maximally planar if adding any missing edge makes it nonplanar. Edge addition and the underlying abstract graph follow Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors. Maximal plane refers to a fixed embedding; maximally planar refers to the abstract graph.
Depends on
- Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs
- Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
Used by
- Every plane triangulation with at least three vertices is connected Lemma
- A two-connected plane graph of order at least three is maximal exactly when every face is triangular Proposition
- For a two-connected planar graph of order at least three, maximal planarity is equivalent to having 3n-6 edges Proposition
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- R. Diestel, Graph Theory, 6th ed., Chapter 4, Section 4.2 (standard reference, not scraped)