ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28

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A pure blockade can have a perfect pattern that is not a cograph

Example

A pure blockade may have a perfect pattern graph without having a cograph pattern graph.

Facts & Assumptions

Given: The path $P_{5}$ on vertices $0, 1, 2, 3, 4$ , and the singleton blocks $B_{i} : = {i - 1}$ for $1 \leq i \leq 5$ .

[L1]

The preceding example shows that $P_{5}$ is perfect but not a cograph (The five-vertex path is perfect but not a cograph).

[L2]

In the pattern graph of a pure blockade, two indices are adjacent exactly when the corresponding two blocks are complete (The pattern graph of a pure blockade).

[L3]

The graph $P_{5}$ has edges exactly $01, 12, 23, 34$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Verification

technique · direct construction

1.1L3given

Because each block $B_{i}$ is a singleton, every pair $(B_{i}, B_{j})$ is either complete or anticomplete according to whether its two vertices are adjacent. Hence $(B_{1}, \dots, B_{5})$ is a pure blockade in the graph $P_{5}$ .

2.1step 1.1L2L3

By [L2], the pattern graph has an edge $i j$ exactly when the vertices $i - 1$ and $j - 1$ are adjacent in $P_{5}$ . Therefore the pattern graph of $(B_{1}, \dots, B_{5})$ is itself $P_{5}$ .

3.1step 2.1L1∎

Step 2.1 and [L1] show that this pure blockade has a perfect pattern that is not a cograph.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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, The Erdos-Hajnal Conjecture - A Survey, Introduction (standard reference, not scraped)