RemarkRemark: Literature-sourcedProof: Not suppliedSession-authored (Fable 5 assisted)‡ sources checked 2026-08-28‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Substituting perfect graphs preserves perfection

Statement

If $H_{1}$ and $H_{2}$ are perfect finite graphs and the substitution $H_{1} [v \to H_{2}]$ is defined, then $H_{1} [v \to H_{2}]$ is perfect.

Remarks

The page uses this only as a recorded preservation theorem. The substitution operation itself is already defined on page 397.

Depends on

Used by

Substituting two α-narrow graphs yields another α-narrow graph Theorem

Dependency tree · two levels

8 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

Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Theorem 5.1 (standard reference, not scraped)