Hinges are defined for reduced hypergraphs, which are hypergraphs where no hyperedge is contained in another. A set of at least two edges <math>H</math> is a hinge if, for every connected component <math>F</math> w.r.t. <math>H</math>, all nodes in <math>F</math> that are also in <math>H</math> are all contained in a single edge of <math>H</math>.
AnA hinge decomposition is based on the correspondence between constraint satisfaction problems and hypergraphs. The hypergraph associated with a problem has the variables of the problem as nodes are the scopes of the constraints as hyperedges. AnA hinge decomposition of a constraint satisfaction problem is a tree whose nodes are some minimal hinges of the hypergraph associated to the problem and such that some other conditions are met. By the correspondence of problems with hypergraphs, a hinge is a set of scopes of constraints, and can therefore be considered as a set of constraints. The additional conditions of the definition of a hinge decomposition are three, of which the first two ensure equivalence of the original problem with the new one. The two conditions for equivalence are: the scope of each constraint is contained in at least one node of the tree, and the subtree induced by a variable of the original problem is connected. The additional condition is that, if two nodes are joined, then they share exactly one constraint, and the scope of this constraint contains all variables shared by the two nodes.
The maximal number of constraints of a node is the same for all hinge decompositions of the same problem. This number is called the ''degree of cyclicity'' of the problem or its hingewidth.
