Note on Maximal Bisection above Tight Lower Bound

In a graph G = (V,E), a bisection (X,Y ) is a partition of V into sets X and Y such that |X| ≤ |Y | ≤ |X|+1. The size of (X,Y ) is the number of edges between X and Y . In the Max Bisection problem we are given a graph G = (V,E) and are required to find a bisection of maximum size. It is not hard to see that ⌈|E|/2⌉ is a tight lower bound on the maximum size of a bisection of G. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (Max-Bisec-ATLB): decide whether a graph G = (V,E) has a bisection of size at least ⌈|E|/2⌉ + k, where k is the parameter. We show that this parameterized problem has a kernel with O(k) vertices and O(k) edges, i.e., every instance of Max-Bisec-ATLB is equivalent to an instance of Max-Bisec-ATLB on a graph with at most O(k) vertices and O(k) edges.