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 ∈ F r , the rank of the adjacency matrix of G is at most r . Using the family F r 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 ) k log ⁡ k ) n O ( 1 ) . We complement our FPT result by designing polynomial kernels for these problems. [ABSTRACT FROM AUTHOR]