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The mixed-block reachability relation on a blockade
Definition
Let be a blockade in a finite graph . Define a relation on the set of blocks of by declaring if either or there is a sequence of blocks
such that every consecutive pair is mixed in the sense of Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs.
Thus two blocks are -related exactly when one can move from one to the other through a chain of mixed block pairs.
Depends on
Used by
- Mixedness of block pairs is not transitive Counterexample
- The quotient blockade obtained from mixed-block reachability Definition
- A vertex may be mixed on a quotient block while pure on each member block 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
- Mixed-block reachability is an equivalence relation Lemma
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
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Section 6 (standard reference, not scraped)