Constraint hypergraph

The hypergraph whose vertices are the variables of a constraint satisfaction problem and whose hypergraph edges are its constraint scopes. Several constraints can have the same scope. An incidence drawing uses extra scope nodes for display; these do not become extra variables of the original problem. Ordinary treewidth of a constraint hypergraph refers to treewidth of its primal graph, equivalently to a tree decomposition covering each whole scope, not to generalized hypergraph width measures.

The graph with one vertex per variable and an edge between distinct variables occurring in a common constraint scope. Each hypergraph edge of the constraint hypergraph becomes a clique. A constraint satisfaction problem consisting only of all-different constraints on a common constraint domain is exactly a proper graph colouring problem on this primal graph.