Recently, the concept of minimal cut sets has been introduced for studying structural fragility and identifying knock-out strategies in biochemical reaction networks. A minimal cut set (MCS) has been defined as a minimal set of reactions whose removal blocks the operation of a chosen objective reaction. In this report the theoretical framework of MCSs is refined and extended increasing the practical applicability significantly. An MCS is now defined as a minimal (irreducible) set of structural interventions (removal of network elements) repressing a certain functionality specified by a deletion task. A deletion task describes unambiguously the flux patterns (or the functionality) to be repressed. It is shown that the MCSs can be computed from the set of target modes, which comprises all elementary modes that exhibit the functionality to be attacked. Since a deletion task can be specified by several Boolean rules, MCSs can now be determined for a large variety of complex deletion problems and may be utilized for inhibiting very special flux patterns. It is additionally shown that the other way around is also possible: the elementary modes belonging to a certain functionality can be computed from the respective set of MCSs. Therefore, elementary modes and MCSs can be seen as dual representations of network functions and both can be converted into each other. Moreover, there exist a strong relationship to minimal hitting sets known from set theory: the MCSs are the minimal hitting sets of the collection of target modes and vice versa. Another generalization introduced herein is that MCSs need not to be restricted to the removal of reactions they may also contain network nodes. In the light of the extended framework of MCSs, applications for assessing, manipulating, and designing metabolic networks in silico are discussed.