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Iterated mixed quotients of an -overlap blockade terminate at a pure blockade
Statement
For a nonempty -overlap blockade, some iterated mixed quotient is a pure blockade.
Facts & Assumptions
Given: A nonempty initial overlap blockade .
Its blocks form a finite nonempty sequence of nonempty sets (Blockades, their length, their width, and their support).
The quotient blocks are the equivalence-class unions of mixed-block reachability (The quotient blockade obtained from mixed-block reachability).
Proof
If is not pure, two distinct blocks are mixed. They lie in one mixed-reachability class, so [F2] merges at least two blocks and strictly decreases the positive integer number of blocks.
Suppose no iterate were pure. Step 1.1 would give an infinite strictly decreasing sequence of positive integers, the successive numbers of blocks.
The set of values of that sequence has a least element by well-ordering, but its successor in the sequence is smaller, a contradiction. Therefore a first pure iterate exists.
Depends on
Used by
- An H₅-overlap class and its terminal quotient Example
- In a special-vertex comb of a co-E-free graph, vertices in other comb blocks remain pure to every H₅-overlap quotient block Lemma
- The pattern of the terminal H₅-overlap quotient is {H₅,co-E}-free Lemma
- A special-vertex comb in a co-E-free graph admits the {H₅,co-E} structural partition Theorem
Dependency tree · two levels
16 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
- Huang, Ju, and Zhou, Erdős-Hajnal beyond the five-vertex path, proof of Lemma 6.4 (standard reference, not scraped)