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.

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$K_3$ contains $P_3$ as a subgraph but not as an induced subgraph

Statement refuted

Every ordinary subgraph copy of $H$ in $G$ is an induced copy.

Facts & Assumptions

Given: The graphs $P_3$ and $K_3$ .

[F1]

Deleting one edge from $K_3$ leaves a spanning subgraph isomorphic to $P_3$ (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).

[F2]

An induced embedding must preserve both adjacency and nonadjacency (Induced embeddings and induced copies of a graph).

Counterexample

technique · direct

1.1

By deleting one triangle edge, $K_3$ contains an ordinary subgraph copy of $P_3$ .

1.2

Any injection from the three vertices of $P_3$ to the three vertices of $K_3$ is bijective, but the two endpoints of $P_3$ are nonadjacent while their images in $K_3$ are adjacent.

2.1

Thus $K_3$ contains no induced $P_3$ , despite containing an ordinary subgraph copy.

step 1.1step 1.2∎

Remarks

Depends on

Used by

Every hereditary graph class is closed under taking arbitrary subgraphs False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 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)

K3K_3K3​ contains P3P_3P3​ as a subgraph but not as an induced subgraph