CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

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 connected components of a graph partition its vertex set and are its maximal connected subgraphs

Statement

The vertex sets $C_{G} (v)$ of the connected components of a graph $G$ are nonempty, cover $V (G)$ , and any two are equal or disjoint. Each induced graph $G [C_{G} (v)]$ is connected and contains every connected subgraph of $G$ that contains $v$ . Consequently the connected components are precisely the maximal connected subgraphs of $G$ .

Facts & Assumptions

Given: A graph $G = (V, E)$ and the reachability classes $C_{G} (v)$ .

[L1]

Reachability is an equivalence relation on $V$ (Being joined by a walk or path is the same equivalence relation on the vertex set).

[F1]

$C_{G} (v)$ is the set of vertices reachable from $v$ , and the connected component at $v$ is the induced graph $G [C_{G} (v)]$ (Connected graphs and connected components defined by the existence of vertex paths, Subgraphs, induced subgraphs and spanning subgraphs).

Proof

technique · direct

1.1

Reflexivity in [L1] gives $v \in C_{G} (v)$ , so every class is nonempty and the classes cover $V$ .

L1F1

1.2

If $C_{G} (u)$ and $C_{G} (v)$ meet at $w$ , then $u$ reaches $w$ and $w$ reaches $v$ by symmetry; transitivity gives that $u$ reaches $v$ , and another use of symmetry and transitivity shows a vertex reaches $u$ exactly when it reaches $v$ . Thus $C_{G} (u) = C_{G} (v)$ , so distinct classes are disjoint.

L1F1

1.3

If $x, y \in C_{G} (v)$ , then $x$ reaches $v$ and $v$ reaches $y$ ; concatenating these paths gives a walk from $x$ to $y$ , and deleting any repeated segments gives a path all of whose vertices remain reachable from $v$ . Hence $G [C_{G} (v)]$ is connected.

L1F1

1.4

If a connected subgraph $H$ contains $v$ , every vertex $x$ of $H$ is joined to $v$ by a path in $H$ , hence also in $G$ , so $x \in C_{G} (v)$ . Thus $H$ is a subgraph of $G [C_{G} (v)]$ .

2.1

Steps 1.1 and 1.2 give the partition assertion, and steps 1.3 and 1.4 say exactly that its induced subgraphs are connected and maximal among connected subgraphs.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

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

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)