Résumé
Given a graph G = (V, E) and an integer k, an edge modification problem for a graph property Pi consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V, E Delta F) satisfies the property Pi. In the Pi edge-completion problem, the set F of edges is constrained to be disjoint from E; in the Pi edge-deletion problem, F is a subset of E; no constraint is imposed on F in the Pi edge-editing problem. A number of optimization problems can be expressed in terms graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if Pi is all hereditary property characterized by a finite set of forbidden induced subgraphs, then the three Pi edge-modification problems are FPT [4]. It was then natural to ask [4] whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively [15]. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths and pointed out that the problem is already open in the case of P-4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for P-i-free and C-i-free edge deletion problems for large enough i.