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This paper deals with the stabilizability of interconnected systems via linear time-invariant (LTI) decentralized controllers. Given a system with some distinct decentralized fixed modes (DFMs), it is desired to find a desirable control structure (in terms of information flow) for it. Since a decentralized controller consists of a number of non-interacting local controllers, the objective here is to establish certain interactions between the local controllers in order to eliminate the undesirable DFMs. The resultant control configuration in the presence of new interactions will be overlapping (as a more general form of decentralized control). In other words, this paper characterizes all the decentralized overlapping control structures with respect to which none of the undesirable modes are fixed. This objective is achieved by translating the knowledge of the system into some bipartite graphs. Then, the notions of minimal sets and maximal subgraphs are introduced, which lead to a simple combinatorial algorithm for solving the underlying problem. The efficacy of the results obtained is demonstrated in an illustrative example.