Parameterized Algorithms for Cluster Vertex Deletion on Degree-4 Graphs and General Graphs

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.