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

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Edgeless graphs form the hereditary class with forbidden basis $\{K_2\}$

Statement

The class of finite edgeless graphs is hereditary, and its minimal forbidden induced-subgraph basis is $\{K_2\}$ .

Facts & Assumptions

Given: The class $\mathcal E$ of finite edgeless graphs.

[L1]

Complete graphs have minimal forbidden basis $\{\overline K_2\}$ (Complete graphs form the hereditary class with forbidden basis $\{\overline K_2\}$ ).

[L2]

Complementation preserves hereditary classes and complements their minimal forbidden bases (Complementation preserves hereditary classes and complements their minimal forbidden bases).

[F1]

Complements of complete graphs are edgeless, and $\overline{\overline K_2}=K_2$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

Verification

technique · direct

1.1

The class $\mathcal E$ is the complement class of the complete graphs.

2.1

Therefore it is hereditary and its basis is obtained by complementing $\overline K_2$ .

step 1.1L1L2

3.1

That complement is $K_2$ , so the basis is $\{K_2\}$ .

step 2.1F1∎

Depends on

Used by

Complementary hereditary classes have complementary forbidden bases Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 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)

Edgeless graphs form the hereditary class with forbidden basis {K2}\{K_2\}{K2​}