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.

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.

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Induced embeddings compose, and the induced-subgraph relation is transitive up to isomorphism

Statement

If $\varphi:H\hookrightarrow_{\mathrm{ind}}G$ and $\psi:G\hookrightarrow_{\mathrm{ind}}K$ , then $\psi\circ\varphi:H\hookrightarrow_{\mathrm{ind}}K$ . Consequently, being an induced subgraph up to isomorphism is transitive.

Facts & Assumptions

Given: Induced embeddings $\varphi:H\to G$ and $\psi:G\to K$ .

[F1]

An induced embedding is injective and preserves adjacency in both directions (Induced embeddings and induced copies of a graph).

[F2]

A composite of injections is injective (Injection, surjection, bijection).

[F3]

Graph isomorphism is compatible with composition (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

The composite $\psi\circ\varphi$ is injective.

1.2

For distinct $x,y\in V(H)$ , one has $xy\in E(H)$ if and only if $\varphi(x)\varphi(y)\in E(G)$ , if and only if $\psi\varphi(x)\psi\varphi(y)\in E(K)$ .

2.1

Hence $\psi\circ\varphi$ is an induced embedding. Replacing induced copies by their isomorphic representatives gives the stated transitivity up to isomorphism.

step 1.1step 1.2F1F3∎

Depends on

Used by

Every induced subgraph of an mathcal F-free graph is mathcal F-free Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 10 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)