In this paper we deal with the problem of decentralized stabilization for linear and time-invariant plants in feedback control configurations that are subject to sparsity constraints. Recent theoretical advances in decentralized control have proved that the class of stabilizing controllers, satisfying a given sparsity constraint admits a convex representation of the Youla-type, provided that the sparsity constraints imposed on the controller are quadratically invariant with respect to the plant. The most useful feature of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the free parameter, which makes this approach suitable for optimal controller design methods based on convex optimization, numerical algorithms. All these procedures rely indispensably on the fact that some decentralized, stabilizing controller is a priori known, while design procedures for such a decentralized controller to initialize the aforementioned optimization schemes are not yet available. This paper provides necessary and sufficient conditions for such a plant to be stabilizable with a decentralized controller. These conditions are given in terms of the existence of a special type of doubly coprime factorization of the plant, which we call the input/output decoupled, doubly coprime factorization. More importantly, the set of all decentralized stabilizing controllers is characterized via the Youla parametrization. The sparsity constraints on the controller are also recast as convex constraints on the Youla parameter.