From a query decomposition of a given width one can build in linear time a pure query decomposition of the same width; this is obtained by replacing the variables of a node that are not in the constraints of the node with other constraints that contain these variables.
From a query decomposition of a given width one can build in logarithmic space a pure query decomposition of the same width. This is obtained by replacing the variables of a node that are not in the constraints of the node with some constraints that contain these variables.