Split Vertex Deletion meets Vertex Cover: New fixed-parameter and exact exponential-time algorithms

Abstract

In the Split Vertex Deletion problem, given a graph G and an integer k, we ask whether one can delete k vertices from the graph G to obtain a split graph (i.e., a graph, whose vertex set can be partitioned into two sets: one inducing a clique and the second one inducing an independent set). In this paper we study exact (exponential-time) and fixed-parameter algorithms for Split Vertex Deletion.•We show that, up to a factor quasipolynomial in k and polynomial in n, the Split Vertex Deletion problem can be solved in the same time as the well-studied Vertex Cover problem. By plugging in the currently best fixed-parameter algorithm for Vertex Cover due to Chen et al. [Theor. Comput. Sci. 411 (40–42) (2010) 3736–3756], we obtain an algorithm that solves Split Vertex Deletion in time O(1.2738kkO(logk)+n3).•We show that all maximal induced split subgraphs of a given n-vertex graph can be listed in O(3n/3nO(logn)) time.To achieve our goals, we prove the following structural result that may be of independent interest: for any graph G we may compute a family P of size nO(logn) containing partitions of V(G) into two parts, such that for any two disjoint sets XC,XI⊆V(G) where G[XC] is a clique and G[XI] is an independent set, there is a partition in P which contains all vertices of XC on one side and all vertices of XI on the other. Moreover, the family P can be enumerated in O(nO(logn)) time.