Abstract
In the Cluster Vertex Deletion problem, we are given a graph G and an integer k, and the goal is to determine whether we can delete at most k vertices from G to make the remaining graph a cluster, i.e., a graph with each connected component being a complete graph. In this paper, we show that Cluster Vertex Deletion can be solved in \(O^*(1.7549^k)\) time, improving the previous result of \(O^*(1.811^k)\). To obtain this result, one crucial step is to show Cluster Vertex Deletion on graphs of maximum degree at most 4 can be solved in \(O^*(1.7485^k)\) time. After this, we know that the graph will always have a vertex of degree at least 5. Then by adopting the previous method of automated generation of searching trees, we can get the result on general graphs.
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Acknowledgments
The author is grateful to all the anonymous reviewers for fruitful and insightful comments to improve the presentation of the paper. The work is supported by the National Natural Science Foundation of China, under the grants 62372095 and 61972070.
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Tian, K., Xiao, M., Yang, B. (2024). Parameterized Algorithms for Cluster Vertex Deletion on Degree-4 Graphs and General Graphs. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14422. Springer, Cham. https://doi.org/10.1007/978-3-031-49190-0_13
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