Super-simple 2-(v; 5; 1) directed designs and their smallest defining sets

In this paper we investigate the spectrum of super-simple 2-$(v,5,1)$ directed designs (or simply super-simple 2-$(v,5,1)$DDs) and also the size of their smallest defining sets. We show that for all $v\equiv1,5\ ({\rm mod}\ 10)$ except $v=5,15$ there exists a super-simple $(v,5,1)DD$. Also for these parameters, except possibly $v=11,91$, there exists a super-simple 2-$(v,5,1)$DD whose smallest defining sets have at least a half of the blocks.Comment: accepted in Australas. J. of Combin. arXiv admin note: text overlap with arXiv:1205.639

Super-simple (v, 4, 2) directed designs and a lower bound for the minimum size of their defining set

In this paper, we show that for all v\pmod 1 (mod 3), there exists a super- simple (v, 4, 2) directed design. Also, we show that for these parameters there exists a super-simple (v, 4, 2) directed design whose each defining set has at least a half of the blocks.Comment: 17 pages. Discrete Applied Mathematics(accepted