# Bounded-Degree Cut is Fixed-Parameter Tractable

In the bounded-degree cut problem, we are given a multigraph G=(V,E), two disjoint vertex subsets A,B⊆ V, two functions u_A, u_B:V→{0,1,…,|E|} on V, and an integer k≥ 0. The task is to determine whether there is a minimal (A,B)-cut (V_A,V_B) of size at most k such that the degree of each vertex v∈ V_A in the induced subgraph G[V_A] is at most u_A(v) and the degree of each vertex v∈ V_B in the induced subgraph G[V_B] is at most u_B(v). In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a 2^18k|G|^O(1)-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems.

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