Extending a technique introduced by Bowditch, Cristopher Cashen and Natasha Makura provide a canonical JSJ decomposition of the free group relative to a finite set of words admitting a topological interpretation. The key object, first considered by Otal, is the so called "decomposition space" of the set of words: the quotient of the boundary of the free group by the equivariant set of lines generated by it. The approach yields some computability results, involving a generalized notion of a Whitehead graph. We will give an account of their work. This is a continuation of an earlier talk in March.