Parameterized algorithms for Module Map problems

Abstract Motivated by applications in the analysis of genetic networks, we introduce and study the NP-hard Module Map problem which has as input a graph  G = ( V , E ) with red and blue edges and an integer  k and asks to transform  G by at most  k edge modifications into a graph  G ′ which has the following properties: the vertex set of  G ′ can be partitioned into so-called clusters such that inside a cluster every pair of vertices is connected by a blue edge and for two distinct clusters  A and  B either all vertices  u ∈ A and  v ∈ B are connected by a red edge or there is no edge between  A and  B . We show that Module Map can be solved in  O ( 3 k ⋅ ( | V | + | E | ) )  time and  O ( 2 k ⋅ | V | 3 ) time, respectively. Furthermore, we show that Module Map admits a kernel with  O ( k 2 ) vertices.