Applying Modular Decomposition to Parameterized Bicluster Editing

Abstract

A graph G is said to be a cluster graph if G is a disjoint union of cliques (complete subgraphs), and a bicluster graph if G is a disjoint union of bicliques (complete bipartite subgraphs). In this work, we study the parameterized version of the NP-hard Bicluster Graph Editing problem, which consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most k modifications are allowed in the edge set of any input graph (Bicluster( k ) Graph Editing problem), this problem is FPT, solvable in O(4 k m) time by applying a search tree algorithm. It is shown an algorithm with O(4 k + n + m) time, which uses a new strategy based on modular decomposition techniques. Furthermore, the same techniques lead to a new form of obtaining a problem kernel with O(k 2) vertices for the Cluster( k ) Graph Editing problem, in O(n +m) time. This problem consists of obtaining a cluster graph by modifying at most k edges in an input graph. A previous FPT algorithm for this problem was presented by Gramm et al. [11]. In their solution, a problem kernel with O(k 2) vertices and O(k 3) edges is built in O(n 3) time.