ExampleConstruction: AI-adaptedVerification: 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.

A fundamental cycle and a fundamental cut in a fixed spanning tree

Statement

Let $G$ have vertices ${1, 2, 3, 4}$ and edges ${12, 23, 34, 14, 13}$ . For the spanning tree $T$ with edges ${12, 23, 34}$ , the edge $14$ has fundamental cycle $1, 2, 3, 4, 1$ . The edge $23$ has fundamental cut ${23, 13, 14}$ .

Facts & Assumptions

Given: The graph $G$ and spanning subgraph $T$ above.

[L1]

An outside edge and its unique tree path form the fundamental cycle (Every edge outside a spanning tree determines a unique fundamental cycle).

[L2]

Deleting a tree edge partitions the tree into two sides, whose crossing edges form the fundamental cut (Every edge of a spanning tree determines a fundamental cut, and every edge crossing it restores a spanning tree).

[F1]

A connected acyclic spanning subgraph is a spanning tree (Spanning trees of a graph).

Verification

technique · direct computation

1.1

The graph $T$ is the path $1, 2, 3, 4$ , so it is a spanning tree. Its unique $1$ - $4$ path together with $14$ is the stated $4$ -cycle.

L1F1

1.2

Deleting $23$ leaves vertex sides ${1, 2}$ and ${3, 4}$ . Exactly $23, 13, 14$ cross between them in $G$ .

2.1

Hence the displayed cycle and cut are the required fundamental objects.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)