## 5.86. connected

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 connected (i.e.,Β there is a path between any pair of vertices of $G$).

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

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 involving a single connected component.

Typical
$|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|>1$
Symmetry

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

Algorithm

A filtering algorithm for the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint is sketched inΒ [Dooms06]. Beside the pruning associated with the fact that the final graph is symmetric, it is based on the fact that all bridges and cut vertices on a path between two vertices that should for sure belong to the final graph should also belong to the final graph.

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 }$$\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi }$$=1$

Graph class
$\mathrm{\pi \pi \pi Ό\pi Ό\pi ΄\pi \pi \pi Έ\pi ²}$

Graph model

PartΒ (A) of FigureΒ 5.86.1 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 5.86.1 gives the final graph associated with the Example slot.