ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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Complementary hereditary classes have complementary forbidden bases

Statement

The class of edgeless graphs is complementary to the class of complete graphs, and their minimal forbidden bases $\{K_2\}$ and $\{\overline K_2\}$ are complementary.

Facts & Assumptions

Given: The classes $\mathcal K$ of complete graphs and $\mathcal E$ of edgeless graphs.

[L1]

$\mathcal B(\mathcal K)=\{\overline K_2\}$ (Complete graphs form the hereditary class with forbidden basis $\{\overline K_2\}$ ).

[L2]

$\mathcal B(\mathcal E)=\{K_2\}$ (Edgeless graphs form the hereditary class with forbidden basis $\{K_2\}$ ).

[L3]

Complementation complements the minimal basis of a hereditary class (Complementation preserves hereditary classes and complements their minimal forbidden bases).

[F1]

A graph belongs to the complement class of $\mathcal K$ exactly when its complement belongs to $\mathcal K$ .

Verification

technique · direct

1.1

Complementing a complete graph gives an edgeless graph and vice versa, so $\mathcal E=\overline{\mathcal K}$ .

1.2

The two basis graphs satisfy $\overline{\overline K_2}=K_2$ .

L1L2

2.1

Thus the displayed bases are complementary, exactly as the general theorem predicts.

step 1.1step 1.2L3∎

Remarks

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: 21 results over 9 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)