  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.
