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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every nonempty hereditary graph class contains the null graph

Statement

Every nonempty hereditary class of finite graphs contains the null graph $K_{0}$ .

Facts & Assumptions

Given: A nonempty hereditary graph class $C$ .

[F1]

Choose $G \in C$ .

[F2]

The induced subgraph $G [\emptyset]$ is the null graph $K_{0}$ (Subgraphs, induced subgraphs and spanning subgraphs, Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F3]

A hereditary class contains every induced subgraph of each member (Hereditary graph classes).

Proof

technique · direct

1.1

Since $G \in C$ , heredity gives $G [\emptyset] \in C$ .

F1F3

2.1

Since $G [\emptyset] = K_{0}$ , the null graph belongs to $C$ .

step 1.1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)