CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

The class of connected graphs is not hereditary

Statement refuted

The class of connected finite graphs is hereditary.

Facts & Assumptions

Given: The path $P_3=v_0v_1v_2$ .

[F1]

$P_3$ is connected (Connected graphs and connected components defined by the existence of vertex paths, Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

[F2]

The induced subgraph on $\{v_0,v_2\}$ has no edge (Subgraphs, induced subgraphs and spanning subgraphs).

[F3]

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

Counterexample

technique · direct

1.1

The graph $P_3$ belongs to the class of connected graphs.

1.2

Its induced subgraph on the endpoints is $\overline K_2$ , which is disconnected.

2.1

Hence this class is not closed under induced subgraphs and is not hereditary.

step 1.1step 1.2F3∎

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: 14 results over 7 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

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)
Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)