  In this article we give necessary and sufficient conditions for two triples
of integers to be realized as the Thurston-Bennequin number and the rotation
number of a Legendrian theta-graph with all cycles unknotted. We show that
these invariants are not enough to determine the Legendrian class of a
topologically planar theta-graph. We define the transverse push-off of a
Legendrian graph and we determine its self linking number for Legendrian
theta-graphs. In the case of topologically planar theta-graphs, we prove that
the topological type of the transverse push-off is that of a pretzel link.
