2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165218We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface.7 pages, 1 figureGeometric Topology57N10; 57M25text