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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Cographs by the singleton, disjoint-union, and complete-connection recursion
Definition
The class of cographs is defined recursively as follows.
- The null graph is a cograph.
- Every one-vertex graph is a cograph.
- If and are vertex-disjoint cographs, then their disjoint union is a cograph.
- If and are vertex-disjoint cographs, then their complete connection is a cograph.
- No graph is a cograph unless it is obtained from the previous clauses by finitely many applications.
Equivalently, a finite graph is a cograph exactly when it is null or can be built from singletons by repeatedly taking disjoint unions and complete connections. In particular, every cograph with at least two vertices has a last construction step in which it is obtained from two nonempty smaller cographs by one of those two operations.
Depends on
Used by
- The four-cycle is a cograph Example
- Every induced subgraph of a cograph is a cograph Lemma
- A pure blockade with a cograph pattern has additive kappa Theorem
- A tau-critical graph has no wide pure blockade with cograph pattern Theorem
- Every cograph is perfect Theorem
- Every nontrivial cograph is disconnected or has disconnected complement Theorem
- The cographs are exactly the P₄-free graphs Theorem
- The Erdos-Hajnal property is equivalent to the large-cograph, large-perfect, and kappa formulations Theorem
Dependency tree · two levels
4 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
- Tero Harju, Lecture Notes on Combinatorial Structures in Graph Theory, Exercise 5.2 (standard reference, not scraped)