LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Complementation swaps cliques with stable sets, so $\omega(\overline G)=\alpha(G)$

Statement

For every finite graph $G$ , a vertex set is a clique in $G$ if and only if it is a stable set in $\overline G$ . Consequently,

$\omega(\overline G)=\alpha(G),\qquad \alpha(\overline G)=\omega(G).$

Facts & Assumptions

Given: A finite graph $G$ and $X\subseteq V(G)$ .

[F1]

A clique has all possible edges among its vertices, while a stable set has none (Cliques, stable sets, the clique number $\omega(G)$ and stability number $\alpha(G)$ ).

[F2]

Distinct vertices are adjacent in $\overline G$ exactly when they are nonadjacent in $G$ (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

Every pair of distinct vertices in $X$ is adjacent in $G$ if and only if no such pair is adjacent in $\overline G$ .

2.1

Thus $X$ is a clique in $G$ if and only if it is stable in $\overline G$ , and symmetrically $X$ is stable in $G$ if and only if it is a clique in $\overline G$ .

step 1.1F1

3.1

The same vertex sets occur in the paired maximizations and retain their cardinalities, so the two displayed equalities follow.

step 2.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 13 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

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)
Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)
ISGCI, Self-complementary classes (standard reference, not scraped)

Complementation swaps cliques with stable sets, so ω(G‾)=α(G)\omega(\overline G)=\alpha(G)ω(G)=α(G)