DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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 complete connection of two disjoint graphs

Definition

Let $G = (V, E)$ and $H = (W, F)$ be finite simple graphs with $V \cap W = \emptyset$ . The complete connection of $G$ and $H$ is the graph

$G \nabla H : = (V \cup W, E \cup F \cup {{v, w} : v \in V, w \in W}) .$

Thus $G \nabla H$ contains $G$ and $H$ on their own vertex sets and adds every possible edge between the two sides. When $X \subseteq V$ and $Y \subseteq W$ , the induced subgraph on $X \cup Y$ is exactly $G [X] \nabla H [Y]$ (Subgraphs, induced subgraphs and spanning subgraphs).

Depends on

Used by

Dependency tree · two levels

3 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)