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

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Complete graphs form the hereditary class with forbidden basis ${{\overline{K}}_{2}}$

Statement

The class of finite complete graphs is hereditary, and its minimal forbidden induced-subgraph basis is ${{\overline{K}}_{2}}$ .

Facts & Assumptions

Given: The class $K$ of finite complete graphs.

[F1]

Every induced subgraph of a complete graph is complete (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A graph is not complete exactly when it has two nonadjacent vertices.

[F3]

A minimal forbidden graph is outside the class while all proper induced subgraphs are inside (Minimal forbidden induced subgraphs and forbidden bases).

[L1]

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

[F4]

Heredity means closure under isomorphism and induced subgraphs (Hereditary graph classes).

Verification

technique · direct

1.1

The class $K$ is isomorphism-closed and closed under induced subgraphs, so it is hereditary.

F1F4

1.2

The graph ${\overline{K}}_{2}$ is not complete, while each of its proper induced subgraphs is $K_{0}$ or $K_{1}$ and is complete. Thus it is minimally forbidden.

1.3

Every noncomplete graph has two nonadjacent vertices, and they induce ${\overline{K}}_{2}$ . Hence avoiding ${\overline{K}}_{2}$ is equivalent to being complete.

2.1

Therefore ${{\overline{K}}_{2}}$ is the unique minimal forbidden basis of $K$ .

step 1.1step 1.2step 1.3L1∎

Depends on

Used by

Dependency tree · two levels

16 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)

Complete graphs form the hereditary class with forbidden basis {K‾2}{K2​}