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

TL;DR

This paper explores the existence and properties of super-simple 2-(v,5,1) directed designs, establishing existence for many parameters and analyzing the size of their smallest defining sets, with some exceptions.

Contribution

It proves the existence of super-simple 2-(v,5,1) directed designs for all v ≡ 1,5 (mod 10) except 5 and 15, and analyzes the size of their smallest defining sets.

Findings

Existence of super-simple 2-(v,5,1)DDs for all v ≡ 1,5 (mod 10) except 5, 15.

Super-simple 2-(v,5,1)DDs with smallest defining sets containing at least half of the blocks.

Possible exceptions for v=11, 91 regarding the size of defining sets.

Abstract

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.