Max-Cut Above Spanning Tree is Fixed-Parameter Tractable

Abstract

Every connected graph on n vertices has a cut of size at least \(n-1\). We call this bound the ‘spanning tree bound’. In the Max-Cut Above Spanning Tree (Max-Cut-AST) problem, we are given a connected n-vertex graph G and a non-negative integer k, and the task is to decide whether G has a cut of size at least \(n-1+k\). We show that Max-Cut-AST admits an algorithm that runs in time \(\mathcal {O}(8^kn^{\mathcal {O}(1)})\), and hence it is fixed parameter tractable with respect to k. Furthermore, we show that Max-Cut-AST has a polynomial kernel of size \(\mathcal {O}(k^5)\).