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(1.2738^k * k^O(log k) + n^O(1)). 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^O(log n) containing partitions of V(G) into two parts, such for any two disjoint subsets X_C, X_I of V(G) where G[X_C] is a clique and G[X_I] is an independent set, there is a partition in P which contains all vertices of X_C on one side and all vertices of X_I on the other.