ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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 $E$ 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

1.1F1

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

2.1step 1.1L1L2

Therefore it is hereditary and its basis is obtained by complementing ${\overline{K}}_{2}$ .

3.1step 2.1F1∎

That complement is $K_{2}$ , so the basis is ${K_{2}}$ .

Depends on

Used by

Complementary hereditary classes have complementary forbidden bases Example

Dependency tree · two levels

12 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)

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