LemmaStatement: AI-adaptedProof: 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.

Deleting any edge of a tree separates it into exactly two tree components

Statement

If $T$ is a tree and $e=uv\in E(T)$ , then $T-e$ has exactly two connected components, each of which is a tree.

Facts & Assumptions

Given: A tree $T$ and an edge $e=uv$ .

[F1]

A tree is connected and has no cycle (Trees, forests, leaves and isolated vertices).

[F2]

$T-e$ has the same vertices and all edges of $T$ except $e$ (Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors).

[F3]

Connectedness is equivalently the existence of a path between every two vertices (Connected graphs and connected components defined by the existence of vertex paths, Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

Proof

technique · direct

1.1

The vertices $u$ and $v$ lie in distinct components of $T-e$ : otherwise a $u$ - $v$ path in $T-e$ together with $e$ would be a cycle in $T$ .

F1F2F3

1.2

Every vertex $x$ lies in the component of $u$ or the component of $v$ . Indeed, take an $x$ - $u$ path in $T$ ; if it avoids $e$ it joins $x$ to $u$ in $T-e$ , and if it uses $e$ then its segment from $x$ to $v$ avoids $e$ .

F1F2F3

2.1

Thus $T-e$ has exactly two components.

step 1.1step 1.2

3.1

Each component is connected by definition and contains no cycle because it is a subgraph of the acyclic graph $T$ . Hence each component is a tree.

F1F3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 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)