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| [[Image:Tree-decomposition-4.svg]]
| Each node of the tree is made a variable. Its ___domain is the set of solutions of the partial problem, which is a set of tuples. The constraints of the new problem allow only the tuples that contain equal values of the original variables. In the figure, equality of <math>y</math> is shown: the corresponding constraint is only met by the first tuple of <math>(x,y,z)</math> and the first tuple of <math>(y,w)</math>, and by the second tuple of <math>(x,y,z)</math> and the second tuple of <math>(y,w)</math>. Moreover, the second tuple of<math> (x,y,z)</math> cannot find a satisfied tuple in its left child (<math> (u,x,z)</math>). Thus, the second tuple of <math>(u,x,z)</math> should be removed.
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