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.

中文翻译：

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最大分割子图和子图及相关类的有效枚举

在本文中，我们对输入任意图 $G$ 并在输出中枚举 $G$ 满足给定属性 $\Pi$ 的所有（包含方式）最大“子图”的算法感兴趣。在整篇论文中，我们研究了几个不同的属性 $\Pi$，所考虑的子图的概念（诱导与否）会因结果而异。更准确地说，我们提出了有效的算法来列出给定输入图的所有最大分割子图、子图和一些图的子类。这里介绍的所有算法都以多项式延迟运行，而且对于分裂图，它只需要多项式空间。为了开发最大分裂（边）子图的算法，我们在最大分裂子图和辅助图的最大独立集之间建立双射。对于 cographs 和一些子类，算法依赖于 Conte & Uno 最近引入的名为 Proximity Search 的框架。最后，我们考虑扩展问题，它包括确定是否存在满足属性 $\Pi$ 的最大诱导子图，该属性包含一组规定的顶点并避免另一组顶点。我们证明这个问题对于每个“有趣的”遗传属性 $\Pi$ 都是 NP 完全的。我们将硬度结果扩展到扩展问题的某些特定边缘版本。这包括决定是否存在满足属性 $\Pi$ 的最大诱导子图，该属性包含一组规定的顶点并避免另一组顶点。我们证明这个问题对于每个“有趣的”遗传属性 $\Pi$ 都是 NP 完全的。我们将硬度结果扩展到扩展问题的某些特定边缘版本。这包括决定是否存在满足属性 $\Pi$ 的最大诱导子图，该属性包含一组规定的顶点并避免另一组顶点。我们证明这个问题对于每个“有趣的”遗传属性 $\Pi$ 都是 NP 完全的。我们将硬度结果扩展到扩展问题的某些特定边缘版本。