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.
Minimal forbidden induced subgraphs and forbidden bases
Definition
Let be a hereditary class. A finite graph is a minimal forbidden induced subgraph for when
but every proper induced subgraph with belongs to (Subgraphs, induced subgraphs and spanning subgraphs).
The minimal forbidden basis is the class of all such graphs, understood up to isomorphism. More generally, a family is a forbidden induced-subgraph basis for when exactly when is -free (-free and -free graphs under the induced-subgraph convention). Minimality here concerns proper induced subgraphs, not ordinary subgraphs (Induced embeddings and induced copies of a graph, Hereditary graph classes).
Depends on
Used by
- Complete graphs form the hereditary class with forbidden basis {overline K₂} Example
- Every hereditary graph class has a finite forbidden induced-subgraph basis False statement
- Complementation preserves hereditary classes and complements their minimal forbidden bases Theorem
- Every hereditary graph class is determined by its unique minimal forbidden induced subgraphs Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 13 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)