## 5.259. orth_link_ori_siz_end

 DESCRIPTION LINKS GRAPH
Origin

Used by several constraints between orthotopes

Constraint

$\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚘𝚛𝚒}_\mathrm{𝚜𝚒𝚣}_\mathrm{𝚎𝚗𝚍}\left(\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}\right)$

Argument
 $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚒}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚜𝚒𝚣}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚎𝚗𝚍}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $|\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}|>0$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}$$\left(2,\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴},\left[\mathrm{𝚘𝚛𝚒},\mathrm{𝚜𝚒𝚣},\mathrm{𝚎𝚗𝚍}\right]\right)$ $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}.\mathrm{𝚜𝚒𝚣}\ge 0$ $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}.\mathrm{𝚘𝚛𝚒}\le \mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}.\mathrm{𝚎𝚗𝚍}$
Purpose

Enforce for each item of the $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$ collection the constraint $\mathrm{𝚘𝚛𝚒}+\mathrm{𝚜𝚒𝚣}=\mathrm{𝚎𝚗𝚍}$.

Example
$\left(\begin{array}{c}〈\mathrm{𝚘𝚛𝚒}-2\mathrm{𝚜𝚒𝚣}-2\mathrm{𝚎𝚗𝚍}-4,\mathrm{𝚘𝚛𝚒}-1\mathrm{𝚜𝚒𝚣}-3\mathrm{𝚎𝚗𝚍}-4〉\hfill \end{array}\right)$

The $\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚘𝚛𝚒}_\mathrm{𝚜𝚒𝚣}_\mathrm{𝚎𝚗𝚍}$ constraint holds since the two items $〈\mathrm{𝚘𝚛𝚒}-2\mathrm{𝚜𝚒𝚣}-2\mathrm{𝚎𝚗𝚍}-4〉$ and $〈\mathrm{𝚘𝚛𝚒}-1\mathrm{𝚜𝚒𝚣}-3\mathrm{𝚎𝚗𝚍}-4〉$ respectively verify the conditions $2+2=4$ and $1+3=4$.

Symmetries
• Items of $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$ are permutable.

• One and the same constant can be added to the $\mathrm{𝚘𝚛𝚒}$ and $\mathrm{𝚎𝚗𝚍}$ attributes of all items of $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$.

• One and the same constant can be added to the $\mathrm{𝚜𝚒𝚣}$ and $\mathrm{𝚎𝚗𝚍}$ attributes of all items of $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$.

Usage

Used in the Arc constraint(s) slot for defining some constraints like $\mathrm{𝚍𝚒𝚏𝚏𝚗}$, $\mathrm{𝚙𝚕𝚊𝚌𝚎}_\mathrm{𝚒𝚗}_\mathrm{𝚙𝚢𝚛𝚊𝚖𝚒𝚍}$ or $\mathrm{𝚘𝚛𝚝𝚑𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}$.

Used in
Keywords
Arc input(s)

$\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$

Arc generator
$\mathrm{𝑆𝐸𝐿𝐹}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎}.\mathrm{𝚘𝚛𝚒}+\mathrm{𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎}.\mathrm{𝚜𝚒𝚣}=\mathrm{𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎}.\mathrm{𝚎𝚗𝚍}$
Graph property(ies)
$\mathrm{𝐍𝐀𝐑𝐂}$$=|\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}|$

Graph model

Parts (A) and (B) of Figure 5.259.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{𝐍𝐀𝐑𝐂}$ graph property, the loops of the final graph are stressed in bold.

##### Figure 5.259.1. Initial and final graph of the $\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚘𝚛𝚒}_\mathrm{𝚜𝚒𝚣}_\mathrm{𝚎𝚗𝚍}$ constraint  (a) (b)
Signature

Since we use the $\mathrm{𝑆𝐸𝐿𝐹}$ arc generator on the $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$ collection the number of arcs of the initial graph is equal to $|\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}|$. Therefore the maximum number of arcs of the final graph is also equal to $|\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}|$. For this reason we can rewrite the graph property $\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}|$ to $\mathrm{𝐍𝐀𝐑𝐂}\ge |\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}|$ and simplify $\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}$ to $\overline{\mathrm{𝐍𝐀𝐑𝐂}}$.