The problem of computing all maximal induced subgraphs of a graph G that have a graph property P, also called the maximal P-subgraphs problem, is considered. This problem is studied for hereditary, connected-hereditary and rooted-hereditary graph properties. The maximal P-subgraphs problem is reduced to restricted versions of this problem by providing algorithms that solve the general problem, assuming that an algorithm for a restricted version is given. The complexity of the algorithms are analyzed in terms of total polynomial time, incremental polynomial time and the complexity class P-enumerable. The general results presented allow simple proofs that the maximal P-subgraphs problem can be solved efficiently (in terms of the input and output) for many different properties.