On the Parameterized Complexity of Computing Graph Bisections

The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published that consider the parameterized complexity of this problem. We show that Bisection is FPT w.r.t. the minimum cut size if there is an optimum bisection that cuts into a given constant number of connected components. Our algorithm applies to the more general Balanced Biseparator problem where vertices need to be removed instead of edges. We prove that this problem is W[1]-hard w.r.t. the minimum cut size and the number of cut out components. For Bisection we further show that no polynomial-size kernels exist for the cut size parameter. In fact, we show this for all parameters that are polynomial in the input size and that do not increase when taking disjoint unions of graphs. We prove fixed-parameter tractability for the distance to constant cliquewidth if we are given the deletion set. This implies fixed-parameter algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set.