## 5.206. k_disjoint

Origin
Constraint

$\mathrm{\pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi ΄\pi \pi }\right)$

Type
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Argument
 $\mathrm{\pi \pi ΄\pi \pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\beta ₯1$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi ΄\pi \pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi \pi ΄\pi \pi }|>1$
Purpose

Given $|\mathrm{\pi \pi ΄\pi \pi }|$ sets of domain variables, the $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint forces that no value is assigned to more than one set.

Example
$\left(β©\mathrm{\pi \pi \pi }-β©1,9,1,5βͺ,\mathrm{\pi \pi \pi }-β©2,7,7,0,6,8βͺ,\mathrm{\pi \pi \pi }-β©4,4,3βͺβͺ\right)$

The $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since:

• The set of values $\left\{1,5,9\right\}$ and $\left\{0,2,6,7,8\right\}$ respectively assigned to the variables of the first and second collections have an empty intersection.

• The set of values $\left\{1,5,9\right\}$ and $\left\{3,4\right\}$ respectively assigned to the variables of the first and third collections have an empty intersection.

• The set of values $\left\{0,2,6,7,8\right\}$ and $\left\{3,4\right\}$ respectively assigned to the variables of the second and third collections have an empty intersection.

Typical
$|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$
Symmetries
• Items of $\mathrm{\pi \pi ΄\pi \pi }$ are permutable.

• Items of $\mathrm{\pi \pi ΄\pi \pi }.\mathrm{\pi \pi \pi }$ are permutable.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be replaced by any value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$.

• All occurrences of two distinct values of $\mathrm{\pi \pi ΄\pi \pi }.\mathrm{\pi \pi \pi }.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a value of $\mathrm{\pi \pi ΄\pi \pi }.\mathrm{\pi \pi \pi }.\mathrm{\pi \pi \pi }$ can be renamed to any unused value.

Arg. properties

Contractible wrt. $\mathrm{\pi \pi ΄\pi \pi }$.

Keywords
Arc input(s)

$\mathrm{\pi \pi ΄\pi \pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\left(<\right)\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi \pi ΄\pi \pi }|*\left(|\mathrm{\pi \pi ΄\pi \pi }|-1\right)/2$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.206.1 respectively show the initial and final graph associated with the Example slot. To each vertex corresponds a collection of variables, while to each arc corresponds a $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint.