Vertex Deletion on Split Graphs: Beyond 4-Hitting Set

TLDR: These problems are “implicit” 4-Hitting Set, and thus admit an algorithm with running time \(\mathcal{O}^\star (3.0755^k)\), a kernel with \(\math Cal O(k^3)\) vertices, and a 4-approximation algorithm, and are studied in the realm of parameterized complexity with respect to the number of vertex deletions k as parameter.

Abstract: In vertex deletion problems on graphs, the task is to find a set of minimum number of vertices whose deletion results in a graph with some specific property. The class of vertex deletion problems contains several classical optimization problems, and has been studied extensively in algorithm design. Recently, there was a study on vertex deletion problems on split graphs. One of the results shown was that transforming a split graph into a block graph and a threshold graph using minimum number of vertex deletions is NP-hard. We call the decision version of these problems as Split to Block Vertex Deletion (SBVD) and Split to Threshold Vertex Deletion (STVD), respectively. In this paper, we study these problems in the realm of parameterized complexity with respect to the number of vertex deletions k as parameter. These problems are “implicit” 4-Hitting Set, and thus admit an algorithm with running time \(\mathcal{O}^\star (3.0755^k)\), a kernel with \(\mathcal{O}(k^3)\) vertices, and a 4-approximation algorithm. In this paper, we exploit the structure of the input graph to obtain a kernel for SBVD with \(\mathcal{O}(k^2)\) vertices and FPT algorithms for SBVD and STVD with running times \({\mathcal O^\star }(2.3028^k)\) and \({\mathcal O^\star }(2.7913^k)\).