Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon surface, i.e., an immersed surface with only ribbon singularities (see Wikipedia).
There is a result (let's assume it here) that $M$ admits a contact cell decomposition, i.e., a finite CW-decomposition with the following properties:
The 1-skeleton is a Legendrian graph
Every 2-cell $D$ satisfies $\operatorname{tw}(\partial D,D)=-1$, where $\operatorname{tw}(\gamma, \Sigma)$ is the twisting number of the Legendrian curve $\gamma$ relative to the surface $\Sigma$
$\xi$ is tight when it is restricted to any 3-cell.