Rank Reduction of Directed Graphs by Vertex and Edge Deletions

In this paper we continue our study of graph modification problems defined by reducing the rank of the adjacency matrix of the given graph, and extend our results from undirected graphs to directed graphs. An instance of a graph modification problem takes as input a graph G, a positive integer k and the objective is to delete k vertices (edges) so that the resulting graph belongs to a particular family, \(\mathcal F\), of graphs. Given a fixed positive integer r, we define \(\mathcal{F}_r\) as the family of directed 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 do algorithmic study, both in classical and parameterized complexity, of the following graph modification problems: \(r\)-Rank Vertex Deletion, \(r\)-Rank Edge Deletion. We first show that both the 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 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. Our main structural result, which is the fulcrum of all our algorithmic results, is that for a fixed integer r the size of any “reduced graph” in \(\mathcal{F}_r\) is upper bounded by \(3^r\). This result is of independent interest and generalizes a similar result of Kotlov and Lovasz regarding reduced undirected graphs of rank r.