Alphabeta MathHow 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.
Every graph with at least four vertices has a nontrivial module
Statement
Every graph with at least four vertices has a nontrivial module.
Facts & Assumptions
Given: The claim above and the four-vertex path .
The four-vertex path has only trivial modules (The four-vertex path has only trivial modules, Modules of a graph, and the trivial modules).
A graph with only trivial modules is prime (Prime graphs: those whose only modules are the trivial ones).
Refutation
The claim asserts that every graph with at least four vertices has some module that is neither empty, nor a singleton, nor the whole vertex set.
The graph has four vertices and, by [L1], has no such module.
So the claim is false. Equivalently, [L2] shows that is a prime graph on four vertices.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
13 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
- M. Habib and C. Paul, A Survey on Algorithmic Aspects of Modular Decomposition, sec. 2.4 (standard reference, not scraped)