Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every class defined by forbidden induced subgraphs is hereditary

Statement

For every family F\mathcal F of finite graphs, the class of all F\mathcal F-free finite graphs is hereditary.

Facts & Assumptions

Given: A family F\mathcal F of finite graphs.

[L1]

Induced subgraphs of an F\mathcal F-free graph remain F\mathcal F-free (Every induced subgraph of an F\mathcal F-free graph is F\mathcal F-free).

[F1]

F\mathcal F-freeness is invariant under graph isomorphism (HH-free and F\mathcal F-free graphs under the induced-subgraph convention).

[F2]

Heredity means closure under isomorphism and induced subgraphs (Hereditary graph classes).

Proof

technique · direct
1.1

The class of F\mathcal F-free graphs is closed under isomorphism because an isomorphism transports every induced copy in both directions.

F1
1.2

It is closed under induced subgraphs by L1.

L1
2.1

These are exactly the two requirements for a hereditary class.

step 1.1step 1.2F2∎

Depends on

Used by

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