Geometric problems defined by constraints can be represented by geometric constraint graphs whose nodes are geometric elements and whose arcs represent geometric constraints. Reduction and decomposition are techniques commonly used to analyze geometric constraint graphs in geometric constraint solving. In this paper we first introduce the concept of {em deficit} of a constraint graph. Then we give a new formalization of the decomposition algorithm due to Owen. This new formalization is based on preserving the deficit rather than on computing triconnected components of the graph and is simpler. Finally we apply tree decompositions to prove that the class of problems solved by the formalizations studied here and other formalizations reported in the literature is the same.