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In this paper, we show that for all v ≡ 1 (mod 3), there exists a supersimple (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.

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph [Formula: see text]. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph [Formula: see text]. Finally, we give some examples to clarify these hyperstructures.

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