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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
- Every P₄-free graph has a clique or stable set of size at least the square root of its order Corollary
- Two large anticonnected components give a complete two-blockade Example
- A complete-or-weakly-sparse blockade yields a complete subblockade or an anticonnected thinning Lemma
- A P₄-free graph on q vertices has a homogeneous set of size at least √q Lemma
- Anticonnected block contraction turns an upside-down comb into a pure blockade Lemma
- Every union of connected components is a module, and so is every union of anticonnected components Lemma
- Small anticonnected components yield a complete blockade Lemma
- A pure blockade with a cograph pattern has additive kappa Theorem
- For every k, the class forbidding Pₖ and Pₖ has the strong Erdős–Hajnal property Theorem
- Gallai's modular decomposition theorem: a graph on at least two vertices is disconnected, or has a disconnected complement, or has a modular partition into its maximal proper modules whose quotient is prime Theorem
- The cographs are exactly the P₄-free graphs Theorem
Dependency tree · two levels
12 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
- Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)