An improved parameterized algorithm for the p-cluster vertex deletion problem

Abstract

In the p-Cluster Vertex Deletion problem, we are given a graph $$G=(V,E)$$G=(V,E) and two parameters k and p, and the goal is to determine if there exists a subset X of at most k vertices such that the removal of X results in a graph consisting of exactly p disjoint maximal cliques. Let $$r=p/k$$r=p/k. In this paper, we design a branching algorithm with time complexity $$O(\alpha ^k+|V||E|)$$O(źk+|V||E|), where $$\alpha $$ź depends on r and has a rough upper bound $$\min \{1.618^{1+r},2\}$$min{1.6181+r,2}. With a more precise analysis, we show that $$\alpha =1.28\cdot 3.57^{r}$$ź=1.28·3.57r for $$r\le 0.219$$r≤0.219; $$\alpha =(1-r)^{r-1}r^{-r}$$ź=(1-r)r-1r-r for $$0.219< r<1/2$$0.219