Reducing Rank of the Adjacency Matrix by Graph Modification

Abstract

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, $\mathcal F$, of graphs. In general the family $\mathcal F$ is defined by forbidden subgraph/minor characterization. In this paper rather than taking a structural route to define $\mathcal F$, we take algebraic route. More formally, given a fixed positive integer r, we define $\mathcal{F}_r$ as the family of graphs where for each $G\in \mathcal{F}_r$, the rank of the adjacency matrix of G is at most r. Using the family $\mathcal{F}_r$ 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 $2^{\mathcal{O}(k \log r)}n^{\mathcal{O}(1)}$ for $r$-Rank Vertex Deletion, and an algorithm for $r$-Rank Edge Deletion and $r$-Rank Editing running in time $2^{\mathcal{O}(f(r) \sqrt{k} \log k )}n^{\mathcal{O}(1)}$. We complement our FPT result by designing polynomial kernels for these problems.