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.

翻译：本文研究输入任意图$G$，并输出所有满足给定性质$\Pi$的（包含意义下）最大“子图”的算法。全文针对多种不同性质$\Pi$展开研究，且所考虑的子图概念（是否诱导）因结果而异。具体而言，我们提出高效算法以列举给定输入图的所有最大分裂子图、子余图及余图的若干子类。本文所有算法均具有多项式延迟，且针对分裂图仅需多项式空间。为开发最大分裂（边）子图的算法，我们建立了最大分裂子图与辅助图最大独立集之间的双射关系。针对余图及其子类，算法基于Conte与Uno近期提出的“邻近搜索”框架。最后考虑扩展问题：判定是否存在满足性质$\Pi$的最大诱导子图，该子图包含指定顶点集且避开另一顶点集。我们证明该问题对任意“有趣”遗传性质$\Pi$均为NP完全问题，并将这一困难性结论推广至扩展问题的特定边版本。