Résumé en anglais

The problem of finding the maximum number of vertex-disjointuni-color paths in anedge-colored graph (called MaxCDP) has been recently introduced in literature, motivatedby applications in social network analysis. In this paper we investigate how the complexity ofthe problem depends on graph parameters (namely the number of vertices to remove to makethe graph a collection of disjoint paths and the size of the vertex cover of the graph), which makes sense since graphs in social networks are not random and have structure. The problemwas known to be hard to approximate in polynomial time and not fixed-parameter tractable (FPT) for the natural parameter. Here, we show that it is still hard to approximate, evenin FPT-time. Finally, we introduce a new variant of the problem, called MaxCDDP, whosegoal is to find the maximum number of vertex-disjoint and color-disjoint uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDPis already hard on graphs at distance two from disjoint paths.