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✓ 4 results · all verified · 4 also independently AI-judged

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Blockades, Combs and Pattern Graphs — Examples

1 · Prerequisites

2 · Summary

These examples separate the different blockade notions that are easy to blur in prose. They record the basic bookkeeping of length and width, show that purity is weaker than being globally complete or globally anticomplete, exhibit a small comb, and make the order dependence of directional sparsity explicit.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: Not suppliedjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

A three-block blockade and its width

Example

Let $G$ be any finite graph containing nine distinct vertices $a, b, c, d, e, f, g, h, i$ . If

$B = ({a, b, c}, {d, e}, {f, g, h, i}),$

then $B$ is a blockade of length $3$ and width $2$ , and

$V (B) = {a, b, c, d, e, f, g, h, i} .$

ExampleConstruction: AI-generatedVerification: Not suppliedjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

A pure blockade that is neither complete nor anticomplete

Example

Take three singleton blocks $B_{1} = {1}$ , $B_{2} = {2}$ , $B_{3} = {3}$ , and let the graph have edges $12$ and $23$ but not $13$ . Then

$(B_{1}, B_{2})$ is complete,
$(B_{2}, B_{3})$ is complete, and
$(B_{1}, B_{3})$ is anticomplete.

So $(B_{1}, B_{2}, B_{3})$ is pure, but it is neither complete nor anticomplete.

ExampleConstruction: AI-generatedVerification: Not suppliedjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

A three-tooth comb

Example

Let $G$ be a graph with six pairwise distinct vertices $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ . Put

$B_{1} = {b_{1}}, B_{2} = {b_{2}}, B_{3} = {b_{3}} .$

If $a_{i}$ is adjacent to $b_{i}$ and to no other $b_{j}$ , then

$((a_{1}, B_{1}), (a_{2}, B_{2}), (a_{3}, B_{3}))$

is a $(3, 1)$ -comb.

ExampleConstruction: AI-generatedVerification: Not suppliedjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

The pattern graph of a pure blockade can be a path

Example

For the pure blockade of A pure blockade that is neither complete nor anticomplete, the pattern graph has vertices ${1, 2, 3}$ and edges $12$ and $23$ but not $13$ . So its pattern graph is the path $P_{3}$ .

ExampleConstruction: AI-generatedVerification: Not suppliedjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

A sparse orientation of four blocks

Example

Let $B = (B_{1}, B_{2}, B_{3}, B_{4})$ with each later block $0$ -sparse to each earlier block. Then directing every pair from the later index to the earlier index gives a sparse orientation of $B$ : each oriented edge points from a block with no cross-neighbors into the block it avoids.

False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

FALSE: every pure blockade is either complete or anticomplete

Statement

False claim: every pure blockade is either complete or anticomplete.

Facts & Assumptions

Given: Three singleton blocks $B_{1} = {1}$ , $B_{2} = {2}$ , $B_{3} = {3}$ with edges $12$ and $23$ but not $13$ .

Proof

technique · direct

1.1

For this blockade, $(B_{1}, B_{2})$ and $(B_{2}, B_{3})$ are complete but $(B_{1}, B_{3})$ is anticomplete, so the blockade is pure.

given

2.1

So the blockade is pure, but not every pair is complete and not every pair is anticomplete. Hence it is neither complete nor anticomplete.

step 1.1

3.1

Therefore the claim is false.

step 2.1∎

False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

FALSE: reversing the order of the blocks never changes $x$ -sparsity

Statement

False claim: the order of the blocks is irrelevant for the property of being $x$ -sparse.

Facts & Assumptions

Given: Two blocks $X = {u}$ and $Y = {y_{1}, y_{2}}$ with the single edge $u y_{1}$ and with $c = \frac{1}{2}$ as the sparsity parameter.

[L1]

An ordered blockade $(B_{1}, B_{2})$ is $c$ -sparse exactly when the later block $B_{2}$ is $c$ -sparse to the earlier block $B_{1}$ (Complete, anticomplete, pure, weakly sparse, and $x$ -sparse blockades).

Proof

technique · direct

1.1

The one-vertex block $X$ is $c$ -sparse to $Y$ , because $u$ has one neighbor in $Y$ and $1 \leq c ∣ Y ∣ = \frac{1}{2} \cdot 2 = 1$ . By [L1], the ordered blockade $(Y, X)$ is therefore $c$ -sparse.

givenL1algebra

1.2

The block $Y$ is not $c$ -sparse to $X$ , because the vertex $y_{1}$ has one neighbor in $X$ but $1 > c ∣ X ∣ = \frac{1}{2}$ . By [L1], the reversed blockade $(X, Y)$ is therefore not $c$ -sparse.

givenL1algebra

2.1

Therefore reversing the order can change $x$ -sparsity, so the claim is false.

step 1.1step 1.2∎

CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

Weak sparsity of a pair does not force directional sparsity

Statement refuted

Refuted claim: if a pair $(X, Y)$ is weakly $c$ -sparse, then $X$ is $c$ -sparse to $Y$ .

Facts & Assumptions

Given: $X = {x_{1}, x_{2}}$ , $Y = {y_{1}, y_{2}}$ , $c = \frac{1}{2}$ , and the edges $x_{1} y_{1}, x_{1} y_{2}$ only.

Counterexample

technique · direct

1.1

The ordered cross-edge count is $e (X, Y) = 2$ , so $e (X, Y) = 2 \leq \frac{1}{2} ∣ X ∣ ∣ Y ∣ = \frac{1}{2} \cdot 2 \cdot 2 = 2$ . Thus $(X, Y)$ is weakly $\frac{1}{2}$ -sparse.

givenalgebra

1.2

But $x_{1}$ has two neighbors in $Y$ , and $2 > \frac{1}{2} ∣ Y ∣ = \frac{1}{2} \cdot 2 = 1$ . So $X$ is not $\frac{1}{2}$ -sparse to $Y$ .

givenalgebra

2.1

Therefore weak sparsity does not imply directional sparsity.

step 1.1step 1.2∎

CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27Open item page →

If a tooth sees a foreign block, the structure is not a comb

Statement refuted

Refuted claim: a sequence of teeth and blocks is still a comb even if some tooth is adjacent to a foreign block.

Facts & Assumptions

Given: Two blocks $B_{1} = {b_{1}}$ and $B_{2} = {b_{2}}$ , and two teeth $a_{1}, a_{2}$ with edges $a_{1} b_{1}$ , $a_{1} b_{2}$ , and $a_{2} b_{2}$ .

Proof

technique · direct

1.1

The tooth $a_{1}$ is complete to its own block $B_{1}$ , but it is also adjacent to the foreign block $B_{2}$ .

given

2.1

The definition Combs in a graph requires every tooth $a_{i}$ to be anticomplete to every foreign block $B_{j}$ with $j \neq i$ . Step 1.1 violates that requirement, so this data is not a comb.

step 1.1given

3.1

Therefore the claim is false.

step 2.1∎

Blockades, Combs and Pattern Graphs — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A three-block blockade and its width

Example

A pure blockade that is neither complete nor anticomplete

Example

A three-tooth comb

Example

The pattern graph of a pure blockade can be a path

Example

A sparse orientation of four blocks

Example

FALSE: every pure blockade is either complete or anticomplete

Statement

Facts & Assumptions

Proof

FALSE: reversing the order of the blocks never changes xx-sparsity

Statement

Facts & Assumptions

Proof

Weak sparsity of a pair does not force directional sparsity

Statement refuted

Facts & Assumptions

Counterexample

If a tooth sees a foreign block, the structure is not a comb

Statement refuted

Facts & Assumptions

Proof

Sources

FALSE: reversing the order of the blocks never changes $x$ -sparsity