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The controllability and the localizability problems are considered under the decentralized information structure using some concepts from graph theory. First of all, the information structure graph is introduced for a decentralized control system based upon local output controllability, matrices between stations when a local output feedback law is applied, and some invariant properties of the graph with respect to local output feedback laws are driven. Then by using these results, necessary and sufficient conditions of controllability under decentralized information structure are obtained. Also obtained are several conditions for localizability, of decentralized control systems, which is newly introduced in the present paper as an extended concept of a class of disturbance localization and decoupling for centralized systems.