ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02

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.

In $K_{2, 3}$ , the two vertices in the two-part have three internally disjoint paths and a minimum separator of size three

Example

Let $K_{2, 3}$ have two-part ${x_{1}, x_{2}}$ and three-part ${y_{1}, y_{2}, y_{3}}$ . The paths $x_{1} y_{i} x_{2}$ for $i = 1, 2, 3$ are internally vertex-disjoint, and ${y_{1}, y_{2}, y_{3}}$ is a minimum $x_{1}$ - $x_{2}$ separator.

Facts & Assumptions

Given: The displayed complete bipartite graph $K_{2, 3}$ .

[L1]

For nonadjacent terminals, finite vertex Menger equates maximum internally disjoint paths and minimum vertex separators (Menger's theorem: the finite directed and undirected arc, edge and nonadjacent-vertex forms).

Verification

Verification technique: direct.

1.1

The three paths $x_{1} y_{1} x_{2}$ , $x_{1} y_{2} x_{2}$ , and $x_{1} y_{3} x_{2}$ have distinct internal vertices.

given

1.2

Deleting all three $y_{i}$ destroys every $x_{1}$ - $x_{2}$ path, while deleting fewer leaves some $y_{i}$ and its two-edge path.

1.3

The terminals are nonadjacent, so [L1] agrees with the direct calculation: both the packing and separator numbers equal three.

2.1

This is a concrete equality case for the local undirected vertex form.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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.

In K2,3K2,3​, the two vertices in the two-part have three internally disjoint paths and a minimum separator of size three