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 \({\mathcal{O}}^{*}(2^{k})\) algorithm(\({\mathcal{O}}^{*}()\) notation hides factors that are polynomial in the input size) improving on the previous best bound of \({\mathcal{O}}^{*} (2.32^{k})\). We also give an \({\mathcal{O}}(k^{3})\)-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 \({\mathcal{O}}^{*}( 2^{ {\mathcal{O}}(\sqrt{k}\log k) } )\) algorithm. We also prove the existence of an \({\mathcal{O}}(k^{2})\) 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): 866893, 2005), and on general graphs.
split graphs parameterized complexity deletion problems subexponential algorithm