Improved Parameterized Algorithms for Cluster Vertex Deletion - Tian, Kangyi; Xiao, Mingyu; Yang, Boting (Theory of computing systems, 2025)
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 graph, i.e., a graph in which every connected component is a complete graph. In this paper, we show that Cluster Vertex Deletion can be solved in time, improving the previous result of . To obtain this result, one crucial step is to show that Cluster Vertex Deletion on graphs of maximum degree at most 4 can be solved in time. For a general graph, after a series of reductions, if the maximum degree of the reduced graph is at most 4, we introduce a new technique, called core branching processing, to solve the problem; if the reduced graph has a vertex of degree at least 5, we adopt the previous method of automated generation of search trees to obtain the improved running time.