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.
Induced embeddings and induced copies of a graph
Definition
Let and be finite simple graphs (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets). An induced embedding of in is an injection such that, for all distinct ,
Thus preserves both adjacency and nonadjacency (Injection, surjection, bijection). Its image is an induced copy of in : the restricted map is an isomorphism from onto that induced subgraph (Subgraphs, induced subgraphs and spanning subgraphs, Graph isomorphisms, automorphisms and graph complements).
We say that is an induced subgraph of up to isomorphism when such an embedding exists.
Depends on
Used by
- K₃ contains P₃ as a subgraph but not as an induced subgraph Counterexample
- H-free and mathcal F-free graphs under the induced-subgraph convention Definition
- Hereditary graph classes Definition
- Minimal forbidden induced subgraphs and forbidden bases Definition
- The induced-embedding count ind_H(G) Definition
- A graph is P₃-free if and only if every connected component is complete Example
- ind_K₂(G)=2|E(G)| under the induced-embedding convention Example
- ind_H(G) is isomorphism-invariant and equals ind_overline H(overline G) Lemma
- Induced embeddings compose, and the induced-subgraph relation is transitive up to isomorphism Lemma
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 20 results over 12 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)