## 5.56. bipartite

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }\right)$

Argument
 $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi },\left[\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi }\right]\right)$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta ₯1$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta €|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi ‘}\right)$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi }\beta ₯1$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi }\beta €|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$
Purpose

Consider a digraph $G$ described by the $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ collection. Select a subset of arcs of $G$ so that the corresponding graph is symmetric (i.e.,Β if there is an arc from $i$ to $j$, there is also an arc from $j$ to $i$) and bipartite (i.e.,Β there is no cycle involving an odd number of vertices).

Example
$\left(\begin{array}{c}β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi \pi \pi \pi }-\left\{2,3\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,4\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,4,5\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi \pi \pi \pi }-\left\{2,3,6\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-5\hfill & \mathrm{\pi \pi \pi \pi }-\left\{3,6\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-6\hfill & \mathrm{\pi \pi \pi \pi }-\left\{4,5\right\}\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since the $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ collection depicts a symmetric graph with no cycle involving an odd number of vertices. The corresponding graph is depicted by FigureΒ 5.56.1.

Typical
$|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|>2$
Symmetry

Items of $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ are permutable.

Algorithm

The sketch of a filtering algorithm for the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint is given inΒ [Dooms06]. Beside enforcing the fact that the graph is symmetric, it checks that the subset of mandatory vertices and arcs is bipartite and removes all potential arcs that would make the previous graph non-bipartite.

Keywords
Arc input(s)

$\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$

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

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi }\right)$
Graph class
 $\beta ’$$\mathrm{\pi \pi \pi Ό\pi Ό\pi ΄\pi \pi \pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$

Graph model

PartΒ (A) of FigureΒ 5.56.2 shows the initial graph from which we start. It is derived from the set associated with each vertex. Each set describes the potential values of the $\mathrm{\pi \pi \pi \pi }$ attribute of a given vertex. PartΒ (B) of FigureΒ 5.56.2 gives the final graph associated with the Example slot.