of the same node. Width 2 guarantees that every consistent partial evaluation can be extended to a solution, but this width is relative to the generated graph. As a result, induced width being 2 is required for strong directional path consistency to guarantee the existence of solutions. Directional i-consistency Directional -consistency is the guarantee that every consistent assignment to variables can be consistently extended to another variable that is higher in the order. Strong directional -consistency is defined in a similar way, but all groups of at most variables are considered. If a problem is strongly directionally -consistent and has width less than and has no empty domain or unsatisfiable constraint, it has solutions. Every problem can be made strongly directionally -consistent, but this operation may increase the width of its corresponding graphs. The constraint propagation procedure that enforces directional consistency is similar to that used for directional arc consistency and path consistency. The variables are considered in turn, from the last to the first according to the order. For variable , the algorithm considers every group of variables that have index lower than and are in a constraint with . Consistency of these variables with is checked and possibly enforced by removing satisfying assignments from the constraint among all these variables (if any, or creating a new one otherwise). This procedure generates a strongly directional -consistent instance. However, it may also add new constraints to the instance. As a result, even if the width of the original problem is , the width of the resulting instance may be greater. If this is the case, directional strong consistency does not imply satisfiability even if no domain is empty and no constraint is unsatisfiable. However, constraint propagation only adds constraints to variables that are lower than the one it is currently considering. As a result, no constraint over a variable is modified or added once the algorithm has dealt with this variable. Instead of considering a fixed , one can modify it to the number of parents of each considered variable (the parents of a variable are the variables of index lower than the variable and that are in a constraint with the variable). This corresponds to considering all parents of a given variables at each step. In other words, for each variable from the last to the first, all its parents are included in a new constraint that limits their