Faster Parameterized Algorithms for Deletion to Split Graphs

An undirected graph is said to be split if its vertex set can be partitioned into two sets such that the subgraph induced on one of them is a complete graph and the subgraph induced on the other is an independent set. We initiate a systematic study of parameterized complexity of the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split. We give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely, 1. for Split Vertex Deletion, the problem of determining whether there are k vertices whose deletion results in a split graph, we give an (Formula presented.) algorithm ((Formula presented.) notation hides factors that are polynomial in the input size) improving on the previous best bound of (Formula presented.). We also give an (Formula presented.)-sized kernel for the problem. 2.For Split Edge Deletion, the problem of determining whether there are k edges whose deletion results in a split graph, we give an (Formula presented.) algorithm. We also prove the existence of an (Formula presented.) kernel. In addition, we note that our algorithm for Split Edge Deletion adds to the small number of subexponential parameterized algorithms not obtained through bidimensionality (Demaine et al. in J. ACM 52(6): 866–893, 2005), and on general graphs.