Planar Legendrian graphs

We prove two results on the classification of trivial Legendrian embeddings $g: G \rightarrow (S^3,\xi_{std})$ of planar graphs. First, the oriented Legendrian ribbon $R_g$ and rotation invariant $\text{rot}_g$ are a complete set of invariants. Second, if $G$ is 3-connected or contains $K_4$ as a minor, then the unique trivial embedding of $G$ is Legendrian simple.