On Legendrian Graphs

We investigate Legendrian graphs in $(\R^3, \xi_{std})$. We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with $tb=-1$ and $rot=0$ if and only if it does not contain $K_4$ as a minor. We show that the pair $(tb, rot)$ does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. For the lollipop graph the pair $(tb,rot)$ determines two Legendrian classes and for the handcuff graph it determines four Legendrian classes.