2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62599We study trivalent graphs in $S^{3}$ whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a simple (i. e. non-loop) level edge.26 pages, 15 figuresGeometric Topology57M25text