On fixed-parameter algorithms for Split Vertex Deletion

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 fixed-parameter algorithms for SPLIT VERTEX DELETION parameterized by k: 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. Plugging the currently best fixed-parameter algorithm for VERTEX COVER due to Chen et al. [TCS 2010], we obtain an algorithm that solves SPLIT VERTEX DELETION in time O(1.2738k k) + n). To achieve our goal, we prove the following structural result that may be of independent interest: for any graph G we may compute a family P of size n n) containing partitions of V (G) into two parts, such 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.