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 Π. All over this paper, we study several different properties Π, 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 [9] called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property Π 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 Π. We extend the hardness result to some specific edge version of the extension problem.

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A dominating set in a directed graph is a set of vertices S such that all the vertices that do not belong to S have an in-neighbour in S. A locating set S is a set of vertices such that all the vertices that do not belong to S are characterized uniquely by the in-neighbours they have in S, i.e. for every two vertices u and v that are not in S, there exists a vertex s ∈ S that dominates exactly one of them. The size of a smallest set of a directed graph D which is both locating and dominating is denoted by γ LD (D). Foucaud, Heydarshahi and Parreau proved that any twin-free digraph D satisfies γ LD (D) ≤ 4n 5 + 1 but conjectured that this bound can be lowered to 2n3 . The conjecture is still open. They also proved that if D is a tournament, i.e. a directed graph where there is one arc between every pair of vertices, then γ LD (D) ≤ ⌈ n 2 ⌉. The main result of this paper is the generalization of this bound to connected local tournaments, i.e. connected digraphs where the in-and out-neighbourhoods of every vertex induce a tournament. We also prove γ LD (D) ≤ 2n 3 for all quasi-twin-free digraphs D that admit a supervising vertex (a vertex from which any vertex is reachable). This class of digraphs generalizes twin-free acyclic graphs, the most general class for which this bound was known.