On defining sets of directed designs

The concept of defining set has been studied in block designs and, under the name critical sets, in Latin squares and Room squares. Here we study defining sets for directed designs. A t-(v, k,'x) directed design (DD) is a pair (V, B), where V is a v-set and B is a collection of ordered blocks (or k-tuples of V), for which each t-tuple of V appears in precisely ,X blocks. A set of blocks which is a subset of a unique t-( v, k, 'x)DD is said to be a defining set of the directed design. As in the case of block designs, finding defining sets seems to be a difficult problem. In this note we introduce some lower bounds for the number of blocks in smallest defining sets in directed designs, determine *The research was partially supported by the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, and by the Australian Research Council Grant A49532477. Australasian Journal of Combinatorics 19(1999), pp.179-190 the precise number of blocks in smallest defining sets for some directed designs with small parameters and point out an open problem relating to the number of blocks needed to define a directed design as compared with the number needed to define its underlying undirected design.