Planar Legendrian $\Theta$-graphs

We classify topologically trivial Legendrian $\Theta$-graphs and identify the complete family of nondestabilizeable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of a $\Theta$-graph or $S^1 \vee S^1$ as a subgraph will have an infinite number of distinct, topologically trivial nondestabilizeable Legendrian embeddings. Additionally, we introduce two moves, vertex stabilization and vertex twist, that change the Legendrian type within a smooth isotopy class.