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 complement of a graph class
Definition
For a graph class , its complement class is
It consists exactly of those graphs whose complements belong to . If is isomorphism-closed, this is equivalently the isomorphism-closed class of complements of members of (Graph isomorphisms, automorphisms and graph complements). The notation does not mean set-theoretic complement inside the class of all graphs.
When is isomorphism-closed, so is , because an isomorphism of graphs induces an isomorphism of their complements.
Depends on
Used by
- Substituting a complete or an edgeless graph for a vertex preserves the Erdős–Hajnal property Corollary
- A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants Proposition
- Complementation preserves hereditary classes and complements their minimal forbidden bases Theorem
Dependency tree · two levels
5 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
- ISGCI, Self-complementary classes (standard reference, not scraped)