### 3.7.9. Alpha-acyclic constraint network(2)

Before defining alpha-acyclic constraint network(2) we first need to introduce the following notions:

• The dual graph of a constraint network $𝒩$ is defined in the following way: to each constraint of $𝒩$ corresponds a vertex in the dual graph and if two constraints have a non-empty set $S$ of shared variables, there is an edge labelled $S$ between their corresponding vertices in the dual graph.

• An edge in the dual graph of a constraint network is redundant if its variables are shared by every edge along an alternative path between the two end points

• If the subgraph resulting from the removal of the redundant edges of the dual graph is a tree the original constraint network is called $\alpha$-acyclic

Alpha-acyclic constraint network(2) denotes an $\alpha$-acyclic constraint network such that, for any pair of constraints, the two sets of involved variables share at most two variables.