TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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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.

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Complementation preserves hereditary classes and complements their minimal forbidden bases

Statement

If $C$ is hereditary, then $\overline{C}$ is hereditary and

$B (\overline{C}) = {\overline{H} : H \in B (C)}$

up to isomorphism.

Facts & Assumptions

Given: A hereditary graph class $C$ .

[F1]

$G \in \overline{C}$ exactly when $\overline{G} \in C$ (The complement of a graph class).

[L1]

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

[F2]

A minimal forbidden graph lies outside the class while all its proper induced subgraphs lie inside (Minimal forbidden induced subgraphs and forbidden bases).

[L2]

A hereditary class is determined by its unique minimal forbidden basis (Every hereditary graph class is determined by its unique minimal forbidden induced subgraphs).

Proof

technique · direct

1.1

Let $G \in \overline{C}$ and $W \subseteq V (G)$ . Then $\overline{G} \in C$ , so $\overline{G} [W] \in C$ by heredity.

1.2

Let $H \in B (C)$ . Then $\overline{H} \notin \overline{C}$ , while for every proper $W ⊊ V (H)$ , $H [W] \in C$ and therefore $\overline{H} [W] = \overline{H [W]} \in \overline{C}$ .

F1F2L1

2.1

Since $\overline{G [W]} = \overline{G} [W]$ , one has $G [W] \in \overline{C}$ . Isomorphism closure is likewise preserved, so $\overline{C}$ is hereditary.

step 1.1L1F1

2.2

Hence $\overline{H} \in B (\overline{C})$ . Applying the same argument to the involution of complementation gives the reverse inclusion.

step 1.2F2

3.1

Therefore the minimal bases are complementary as claimed.

step 2.2L2∎

Depends on

Used by

Dependency tree · two levels

13 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)