Abstract: We present an analysis of the deficiency indices of the 2-D Dirac Operator with Lorentz-scalar interactions supported on a star-graph, with different interaction strengths allowed on different leads. In the general case, we separate variables, decompose the Dirac operator into an orthogonal sum and find that the deficiency indices depend on the number of eigenvalues of the so-called spin-orbit operator within an interval. For the simpler cases when there are two or three leads much more can be said. Examples when the deficiency indices are (2,2) and when the spin-orbit operator has eigenvalues of multiplicity two are included. It is also shown that there is a distinguished self-adjoint extension whose domain lies in H^{1/2}.