A graph is called a split graph if its vertex set can be partitioned into two disjoint sets and such that is a clique and is an independent set. A graph is called a block graph if every biconnected component is a clique. In the Split to Block Vertex Deletion (SBVD) problem the input is a split graph and an integer , and the goal is to decide whether there is a set of at most vertices such that is a block graph. The SBVD problem was shown to be NP-hard by Cao et al. . Choudhary et al.  gave a parameterized algorithm for SBVD whose running time is . In this paper we give an algorithm for SBVD whose running time is .
2 The algorithm
In the 3-Hitting Set problem the input is a family of subsets of a set and an integer , and the goal is to decide whether there is a set of size at most such that for every .
Let be a split graph with a partition of its vertices. Suppose that is a block graph for some set of vertices . If there is a vertex with at least two neighbors in then is adjacent to all the vertices of , otherwise the biconnected component in that contains satisfies and thus this component is not a clique. Moreover, there can be at most one vertex in with at least two neighbors in (if have at least two neighbors in then are in the same biconnected component and thus this component is not a clique). Denote this vertex, if it exists, by . For every we have that has at most one neighbor in .
Therefore, the algorithm for SBVD goes over all possible choice for the vertex . Additionally, the algorithm also inspects the case in which no such vertex exist. For every choice of , the algorithm deletes from all the vertices of that are not adjacent to and decreases the value of by the number of vertices deleted. When the algorithm inspects the case when does not exists, the graph is not modified.
For every choice of , the algorithm generates an instance of 3-Hitting Set as follows: For every vertex that has at least two neighbors, and for every two neighbors of , construct a set . The instance of 3-Hitting Set is where consists of all the sets defined above. Now, the algorithm uses the algorithm of Wahlström  to solve the instance in time. If is a yes instance of 3-Hitting Set then the algorithm return yes. If all the constructed 3-Hitting Set instances, for all choices of , are no instances, the algorithm returns no.
-  Yixin Cao, Yuping Ke, Yota Otachi, and Jie You. Vertex deletion problems on chordal graphs. Theoretical Computer Science, 745:75–86, 2018.
-  Pratibha Choudhary, Pallavi Jain, R Krithika, and Vibha Sahlot. Vertex deletion on split graphs: Beyond 4-hitting set. In International Conference on Algorithms and Complexity (CIAC), pages 161–173, 2019.
-  Magnus Wahlström. Algorithms, measures and upper bounds for satisfiability and related problems. PhD thesis, Department of Computer and Information Science, Linköpings universitet, 2007.