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.
The plane dual multigraph, with a vertex for each face and one crossing edge for each primal edge
Definition
Let be a connected plane graph with face set (Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs). Its dual multigraph has vertex set and one edge for every primal edge . If the two local sides of belong to faces and , then has endpoints .
By Face frontiers are unions of whole edges; a cycle edge borders two faces and a bridge borders one, a bridge has the same face on both local sides, so its dual edge is a loop. Different primal edges may have the same incident face pair, so their duals may be parallel. These are permitted by Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph. The definition is attached to the fixed plane embedding: different embeddings of the same abstract planar graph may have nonisomorphic dual multigraphs.
Depends on
- Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs
- Face frontiers are unions of whole edges; a cycle edge borders two faces and a bridge borders one
- Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph
Used by
Dependency tree · two levels
12 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 4.6 (standard reference, not scraped)
- J. Erickson, Planar Graphs, Section 9 (standard reference, not scraped)