2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74964Let G be a graph in a 3-manifold M. We compress the pair (M,G) along admissible 2-spheres as long as possible. What we get is a root of (M,G). Our main result is that for any pair (M,G) the root exists and is unique. As a corollary we get an easy proof of Petronio's theorem on prime decompositions of 3-orbifolds.15 pages, 3 figuresGeometric Topology57M27text