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.

Complementation swaps cliques with stable sets, so $ω (\overline{G}) = α (G)$

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,

$ω (\overline{G}) = α (G), α (\overline{G}) = ω (G) .$

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 $ω (G)$ and stability number $α (G)$ ).

[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

Dependency tree · two levels

10 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

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)ω(G)=α(G)