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.
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Distinct connected components are anticomplete, and distinct anticonnected components are complete
Statement
Distinct connected components of a graph are anticomplete to one another. Distinct anticomponents are complete to one another.
Facts & Assumptions
Given: A finite graph .
Connected components partition the vertices into maximal connected parts (The connected components of a graph partition its vertex set and are its maximal connected subgraphs).
Anticomponents of are connected components of (The anticonnected components of are exactly the connected components of ).
Complementation swaps anticomplete pairs with complete pairs (Purity is symmetric; complementation swaps complete and anticomplete pairs and preserves mixed pairs).
Anticomplete and complete pairs have respectively no and all cross edges (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).
Proof
No edge joins two distinct connected components, since such an edge would connect them into one component. Thus distinct components are anticomplete.
Distinct anticomponents of are distinct connected components of , so they are anticomplete in by step 1.1 applied there.
Complementing back makes those two sets complete in .
Depends on
- Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs
- The connected components of a graph partition its vertex set and are its maximal connected subgraphs
- The anticonnected components of $G$ are exactly the connected components of $\overline G$
- Purity is symmetric; complementation swaps complete and anticomplete pairs and preserves mixed pairs
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 18 results over 10 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
- Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)