CorollaryStatement: 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.

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

Statement

For finite graphs $G$ and $H$ ,

$G\text{ is }H\text{-free}\quad\Longleftrightarrow\quad \overline G\text{ is }\overline H\text{-free}.$

Facts & Assumptions

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

[F1]

$H$ -free means containing no induced copy of $H$ ( $H$ -free and $\mathcal F$ -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 $H\cong G[W]$ to isomorphisms $\overline H\cong\overline G[W]$ (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

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

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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 8 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

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)

GGG is HHH-free if and only if G‾\overline GG is H‾\overline HH-free