  In this paper, we are interested in algorithms that take in input an
arbitrary graph $G$, and that enumerate in output all the (inclusion-wise)
maximal "subgraphs" of $G$ which fulfil a given property $\Pi$. All over this
paper, we study several different properties $\Pi$, and the notion of subgraph
under consideration (induced or not) will vary from a result to another.
  More precisely, we present efficient algorithms to list all maximal split
subgraphs, sub-cographs and some subclasses of cographs of a given input graph.
All the algorithms presented here run in polynomial delay, and moreover for
split graphs it only requires polynomial space. In order to develop an
algorithm for maximal split (edge-)subgraphs, we establish a bijection between
the maximal split subgraphs and the maximal independent sets of an auxiliary
graph. For cographs and some subclasses , the algorithms rely on a framework
recently introduced by Conte & Uno called Proximity Search. Finally we consider
the extension problem, which consists in deciding if there exists a maximal
induced subgraph satisfying a property $\Pi$ that contains a set of prescribed
vertices and that avoids another set of vertices. We show that this problem is
NP-complete for every "interesting" hereditary property $\Pi$. We extend the
hardness result to some specific edge version of the extension problem.
