Quotient Blockades and Mixing Relations — Examples

The pair $(A,B)$ is mixed: it has edges $a_1{b}_1$ and $a_2{b}_2$, but also nonedges $a_1{b}_2$ and $a_2{b}_1$. The same calculation shows that $(B,C)$ is mixed.

By construction there are no edges between $A$ and $C$, so $(A,C)$ is anticomplete and therefore not mixed.

Thus mixedness can hold for $(A,B)$ and $(B,C)$ while failing for $(A,C)$. This is why [L2] passes to the reachability closure $\mathcal{M}$ rather than treating mixedness itself as an equivalence relation.

Because $(A,B)$ and $(B,C)$ are mixed, there is a mixed chain from $A$ to $C$ through $B$. Thus $A$, $B$, and $C$ lie in the same $\mathcal{M}$-class.

No pair involving $D$ is mixed, so there is no mixed chain from $D$ to any of $A$, $B$, or $C$. Therefore $D$ lies in a different $\mathcal{M}$-class.

By [L1], the quotient blockade has one block equal to the union $A\cup B\cup C$ of the first class and one block equal to $D$ from the second class.

Since $A_1$ and $A_2$ are mixed, [L1] places them in the same mixed-reachability class. They are the only blocks of $\mathcal{L}$, so their class has union $D=A_1\cup A_2$, the unique quotient block.

The vertex $u$ is adjacent to every vertex of $A_1$ and to no vertex of $A_2$. Therefore $u$ is neither complete nor anticomplete to $D$, so $u$ is mixed on $D$. At the same time it is pure to $A_1$ and pure to $A_2$ individually.

This is exactly the phenomenon isolated by [L2]: the mixing appears only after passing to the quotient union.

By item 2 of the configuration and [L1], the two original blocks $A_1,A_2$ form one quotient block $D_1=A_1\cup A_2$, while $D_2=B$ is the other quotient block. Because $B$ is complete to $A_1$ and anticomplete to $A_2$, the quotient blocks $D_1$ and $D_2$ are mixed.

Item 1 of the configuration says that every original block of the blockade is connected, so the connectivity hypothesis of [L2] holds. No vertex of $D_1$ is mixed on $D_2$: every vertex of $A_1$ is complete to $B$, and every vertex of $A_2$ is anticomplete to $B$. The outside vertices satisfy the remaining hypotheses of [L2]: $x$ and $y$ are nonadjacent and complete to $D_1\cup D_2$, while $u\in N(x)\setminus N(y)$ is complete to $D_1$ and anticomplete to $D_2$.

Applying [L2] therefore yields mixed original blocks $A_1,A_2$ inside $D_1$ and vertices $x^{\mathrm{\prime }},y^{\mathrm{\prime }},u^{\mathrm{\prime }}$ outside $A_1\cup A_2$ with the required adjacency pattern. Thus the quotient-level witness has been pushed down to a witness on the original mixed pair inside $D_1$.