LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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

Statement

If $φ : H ↪_{i n d} G$ and $ψ : G ↪_{i n d} K$ , then $ψ \circ φ : H ↪_{i n d} K$ . Consequently, being an induced subgraph up to isomorphism is transitive.

Facts & Assumptions

Given: Induced embeddings $φ : H \to G$ and $ψ : G \to K$ .

[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 $ψ \circ φ$ is injective.

1.2

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

2.1

Hence $ψ \circ φ$ 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

Dependency tree · two levels

7 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

Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)