We construct an efficient model for graphs of finitely generated subgroups of free groups. Using this we give a very short proof of Dicks’s reformulation of the strengthened Hanna Neumann Conjecture as the Amalgamated Graph Conjecture. In addition, we answer a question of Culler and Shalen on ranks of intersections in free groups. The latter has also been done independently by R P Kent IV.
"Graphs of subgroups of free groups." Algebr. Geom. Topol. 9 (1) 327 - 335, 2009. https://doi.org/10.2140/agt.2009.9.327