TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every cograph is perfect

Statement

Every cograph is perfect.

Facts & Assumptions

Given: A cograph $G$ .

[L1]

A cograph is either null or is built from one-vertex graphs by finitely many disjoint unions and complete connections, and every cograph with at least two vertices has a final step of one of those two kinds (Cographs by the singleton, disjoint-union, and complete-connection recursion).

[L2]

A disjoint union of two perfect graphs is perfect (A disjoint union of two perfect graphs is perfect).

[L3]

A complete connection of two perfect graphs is perfect (A complete connection of two perfect graphs is perfect).

[F1]

The null graph and every one-vertex graph are perfect (Perfect graphs).

Proof

technique · induction on a cograph construction of $G$

1.1

We argue by induction on the recursive definition of cographs. If $G$ is the null graph or a one-vertex graph, then [F1] shows that $G$ is perfect.

givenF1base

1.2

Assume now that $G$ is nontrivial and that the claim is already known for the two smaller cographs in its final construction step. By [L1], there exist smaller cographs $G_{1}$ and $G_{2}$ such that $G$ is either $G_{1} ⊔ G_{2}$ or $G_{1} \nabla G_{2}$ . The induction hypothesis makes both $G_{1}$ and $G_{2}$ perfect.

L1ih

2.1

In the disjoint-union case, [L2] shows that $G$ is perfect. In the complete-connection case, [L3] shows that $G$ is perfect.

step 1.2L2L3

3.1

Steps 1.1 and 2.1 close the induction. Therefore every cograph is perfect.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

The Erdos-Hajnal property is equivalent to the large-cograph, large-perfect, and kappa formulations Theorem

Dependency tree · two levels

9 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, The Erdos-Hajnal Conjecture - A Survey, Introduction (standard reference, not scraped)
Tero Harju, Lecture Notes on Combinatorial Structures in Graph Theory, Exercise 5.2 (standard reference, not scraped)