Reducing rank of the adjacency matrix by graph modification

The main topic of this article is to study a class of graph modification problems. A typical graph modification problem takes as input a graph G, a positive integer k and the objective is to add/delete k vertices (edges) so that the resulting graph belongs to a particular family, F, of graphs. In general the family F is defined by forbidden subgraph/minor characterization. In this paper rather than taking a structural route to define F, we take algebraic route. More formally, given a fixed positive integer r, we define F-r, as the family of graphs where for each G is an element of (F)r, the rank of the adjacency matrix of G is at most r. Using the family Fr we initiate algorithmic study, both in classical and parameterized complexity, of following graph modification problems: r-RANK VERTEX DELETION, r-RANK EDGE DELETION and r-RANK EDITING. These problems generalize the classical VERTEX COVER problem and a variant of the d-CLUSTER EDITING problem. We first show that all the three problems are NP-Complete. Then we show that these problems are fixed parameter tractable (FPT) by designing an algorithm with running time 2(O(k log r))n(O(1)) for r-RANK VERTEX DELETION, and an algorithm for r-RANK EDGE DELETION and r-RANK EDITING running in time 2(O(f(r)root k log k))n(O(1)). We complement our FPT result by designing polynomial kernels for these problems. (C) 2016 Elsevier B.V. All rights reserved.