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A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) issuper-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is,any two blocks of the $BIBD$ intersect in at most two points. A$2-(v,k,\lambda)DD$ is simple if its underlying $2-(v,k,2\lambda)BIBD$ issimple, that is, it has no repeated blocks.  A set of blocks which is a subset of a unique $2-(v,k,\lambda)DD$ is said tobe a defining set of the directed design. A smallest defining set, is adefining set which has smallest cardinality. In this paper simultaneously weshow that the necessary and sufficient condition for the existence of asuper-simple $2-(v,4,1)DD$ is $v\equiv1\ ({\rm mod}\ 3)$ and for these valuesexcept $v=7$, there exists a super-simple $2-(v,4,1)DD$ whose smallest definingsets have at least a half of the blocks. And also for all $\epsilon > 0$ thereexists $v_0(\epsilon)$ such that for all admissible $v>v_0$ there exists a$2-(v,4,1)DD$ whose smallest defining sets have at least $(5/8-\frac{c}{v})\mid\mathcal{B}\mid$ blocks, for suitable positive constant c.

Smallest defining sets of super-simple 2 - (v, 4,1) directed designs