5.72. connected

DESCRIPTIONLINKSGRAPH
Origin

[Dooms06]

Constraint

πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš(π™½π™Ύπ™³π™΄πš‚)

Argument
π™½π™Ύπ™³π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšœπšŸπšŠπš›)
Restrictions
πš›πšŽπššπšžπš’πš›πšŽπš(π™½π™Ύπ™³π™΄πš‚,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌])
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰₯1
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰€|π™½π™Ύπ™³π™΄πš‚|
πšπš’πšœπšπš’πš—πšŒπš(π™½π™Ύπ™³π™΄πš‚,πš’πš—πšπšŽπš‘)
π™½π™Ύπ™³π™΄πš‚.𝚜𝚞𝚌𝚌β‰₯1
π™½π™Ύπ™³π™΄πš‚.πšœπšžπšŒπšŒβ‰€|π™½π™Ύπ™³π™΄πš‚|
Purpose

Consider a digraph G described by the π™½π™Ύπ™³π™΄πš‚ 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
πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-{1,2,3},πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-{1,3},πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-{1,2,4},πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-{3,5,6},πš’πš—πšπšŽπš‘-5𝚜𝚞𝚌𝚌-{4},πš’πš—πšπšŽπš‘-6𝚜𝚞𝚌𝚌-{4}

The πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš constraint holds since the π™½π™Ύπ™³π™΄πš‚ collection depicts a symmetric graph involving one single connected component.

Typical
|π™½π™Ύπ™³π™΄πš‚|>1
Symmetry

Items of π™½π™Ύπ™³π™΄πš‚ are permutable.

Algorithm

A filtering algorithm for the πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš 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 cutvertices on a path between two vertices that should for sure belong to the final graph should also belong to the final graph.

See also

common keyword: πšœπš’πš–πš–πšŽπšπš›πš’πšŒΒ (symmetric).

implies: πšœπšπš›πš˜πš—πšπš•πš’_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš.

used in graph description: πš’πš—_𝚜𝚎𝚝.

Keywords

constraint arguments: constraint involving set variables.

constraint type: graph constraint.

final graph structure: connected component, symmetric.

Arc input(s)

π™½π™Ύπ™³π™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš—πš˜πšπšŽπšœ1,πš—πš˜πšπšŽπšœ2)

Arc arity
Arc constraint(s)
πš’πš—_𝚜𝚎𝚝(πš—πš˜πšπšŽπšœ2.πš’πš—πšπšŽπš‘,πš—πš˜πšπšŽπšœ1.𝚜𝚞𝚌𝚌)
Graph property(ies)
𝐍𝐂𝐂=1

Graph class
πš‚πšˆπ™Όπ™Όπ™΄πšƒπšπ™Έπ™²

Graph model

Part (A) of Figure 5.72.1 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 𝚜𝚞𝚌𝚌 attribute of a given vertex. Part (B) of Figure 5.72.1 gives the final graph associated with the Example slot.

Figure 5.72.1. Initial and final graph of the πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš set constraint
ctrs/connectedActrs/connectedB
(a) (b)