Abstract
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 study the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split.We initiate a systematic study and give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely,
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for Split Vertex Deletion, the problem of determining whether there are k vertices whose deletion results in a split graph, we give an \({\cal O}^*(2^k)\) algorithm improving on the previous best bound of \({\cal O}^*({2.32^k})\). We also give an \({\cal O}(k^3)\)-sized kernel for the problem.
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For Split Edge Deletion, the problem of determining whether there are k edges whose deletion results in a split graph, we give an \({\cal O}^*( 2^{ O(\sqrt{k}\log k) } )\) algorithm. We also prove the existence of an \({\cal O}(k^2)\) 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, and on general graphs.
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Ghosh, E. et al. (2012). Faster Parameterized Algorithms for Deletion to Split Graphs. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_10
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