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Separating systems and oriented graphs of diameter two
Journal of Combinatorial Theory Series B
European Journal of Combinatorics
General theory of information transfer: Updated
Discrete Applied Mathematics
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The following model is considered. There is exactly one unknown element in the n-element set. A question is a partition of S into three classes: (A,L,B). If x∈A then the answer is "yes" (or 1), if x∈B then the answer is "no" (or 0), finally if x∈L then the answer can be either "yes" or "no". In other words, if the answer "yes" is obtained then we know that x∈A∪L while in the case of "no" answer the conclusion is x∈B∪L. The mathematical problem is to minimize the minimum number of questions under certain assumptions on the sizes of A,B and L. This problem has been solved under the condition |L|≥k by the author and Krisztián Tichler in previous papers for both the adaptive and non-adaptive cases. In this paper we suggest to solve the problem under the conditions |A|≤a, |B|≤b. We exhibit some partial results for both the adaptive and non-adaptive cases. We also show that the problem is closely related to some known combinatorial problems. Let us mention that the case b=n−a has been more or less solved in earlier papers.