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The pattern graph of a pure blockade
Definition
Let be a pure blockade in a graph . Its pattern graph is the graph with vertex set in which and are adjacent exactly when is complete to .
Because the blockade is pure, every unordered pair of distinct blocks is either complete or anticomplete, so this graph is well defined. A pattern graph is called -free when it contains no induced four-vertex path.
Depends on
Used by
- Omitting cross-block purity breaks the transversal conclusion Counterexample
- The structural comb-partition hypothesis Definition
- A pure blockade can have a perfect pattern that is not a cograph Example
- A two-block pure blockade can realize equality in the additive kappa theorem Example
- The pattern graph of a pure blockade can be a path Example
- A maximal pure blockade with large total a-mass must already have at least ε⁻² blocks Lemma
- Homogeneous sets in pure-blockade patterns lift to complete or anticomplete blockades Lemma
- Pure blockades with P₄-free patterns contain complete or anticomplete subblockades of square-root length Lemma
- The pattern of the terminal H₅-overlap quotient is {H₅,co-E}-free Lemma
- The terminal E overlap pattern is E-free Lemma
- A long blockade yields a wide cograph-pattern subblockade or a rainbow forest Theorem
- A pure blockade with a cograph pattern has additive kappa Theorem
- A pure blockade with a perfect pattern has a large complete or anticomplete subblockade Theorem
- A tau-critical graph has no wide pure blockade with cograph pattern Theorem
- The special-vertex-local structural-partition criterion implies property (*) Theorem
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
- Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdős-Hajnal for graphs with no 5-hole (standard reference, not scraped)