CorollaryStatement: 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.

$G$ is $H$ -free if and only if $\overline{G}$ is $\overline{H}$ -free

Statement

For finite graphs $G$ and $H$ ,

$G is H -free ⟺ \overline{G} is \overline{H} -free .$

Facts & Assumptions

Given: Finite graphs $G$ and $H$ .

[F1]

$H$ -free means containing no induced copy of $H$ ( $H$ -free and $F$ -free graphs under the induced-subgraph convention).

[L1]

Complementation commutes with induced subgraphs ( $\overline{G [W]} = \overline{G} [W]$ for every vertex set $W$ ).

[F2]

Complementation carries isomorphisms $H ≅ G [W]$ to isomorphisms $\overline{H} ≅ \overline{G} [W]$ (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

For every $W \subseteq V (G)$ , one has $G [W] ≅ H$ if and only if $\overline{G} [W] = \overline{G [W]} ≅ \overline{H}$ .

L1F2

2.1

Thus $G$ contains an induced $H$ if and only if $\overline{G}$ contains an induced $\overline{H}$ . Negating both sides gives the claimed equivalence.

step 1.1F1∎

Depends on

Used by

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

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)

GG is HH-free if and only if G‾G is H‾H-free