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The quotient blockade obtained from mixed-block reachability
Definition
Let be a blockade, and let be its mixed-block reachability relation. The quotient blockade is obtained by replacing each -equivalence class of original blocks by its union.
Concretely, if are the -classes, ordered by the least original index of a block they contain, then
Each is called a quotient block. Because the original blocks are pairwise disjoint and each equivalence class is nonempty, the quotient blocks are again pairwise disjoint and nonempty.
Depends on
Used by
- A mixed chain of blocks collapses to one quotient block Example
- A vertex may be mixed on a quotient block while pure on each member block Example
- The quotient-witness reduction in a three-block configuration Example
- A quotient block of connected or anticonnected blocks is again connected or anticonnected Lemma
- A vertex mixed on a quotient block but pure on each member block yields two mixed member blocks with opposite adjacency Lemma
- Blocks from distinct mixed-block classes are pure to each other Lemma
Dependency tree · two levels
5 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
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Section 6 (standard reference, not scraped)