Maximum Balanced Subgraph Problem Parameterized above Lower Bound

We consider graphs without loops or parallel edges in which every edge is assigned + or −. Such a signed graph is balanced if its vertex set can be partitioned into parts V1 and V2 such that all edges between vertices in the same part have sign + and all edges between vertices of different parts have sign − (one of the parts may be empty). It is well-known that every connected signed graph with n vertices and m edges has a balanced subgraph with at least m 2 + n−1 4 edges and this bound is tight. We consider the following parameterized problem: given a connected signed graph G with n vertices and m edges, decide whether G has a balanced subgraph with at least m 2