# Finding cuts of bounded degree: complexity, FPT and exact algorithms, and kernelization

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The MATCHING CUT problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call d-CUT: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the MATCHING CUT problem. Namely, we begin with an NP-hardness reduction for d-CUT on (2d+2)-regular graphs and a polynomial algorithm for graphs of maximum degree at most d+2. The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for MATCHING CUT. Our main technical contribution, building on the techniques of Komusiewicz et al. [IPEC 2018], is a polynomial kernel for d-CUT for every positive integer d, parameterized by the distance to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O^*(2^n).

• 11 publications
• 29 publications
research
01/18/2021

### An FPT algorithm for Matching Cut

In an undirected graph, a matching cut is an edge cut which is also a ma...
research
04/27/2021

### Minimum Stable Cut and Treewidth

A stable cut of a graph is a cut whose weight cannot be increased by cha...
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11/17/2017

### An O^*(1.84^k) Parameterized Algorithm for the Multiterminal Cut Problem

We study the multiterminal cut problem, which, given an n-vertex graph w...
research
08/02/2023

### Polynomial-delay Enumeration Kernelizations for Cuts of Bounded Degree

Enumeration kernelization was first proposed by Creignou et al. [TOCS 20...
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12/06/2021

### Faster Cut Sparsification of Weighted Graphs

A cut sparsifier is a reweighted subgraph that maintains the weights of ...
research
10/08/2019

### Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

In the presented paper we study the Length-Bounded Cut problem for speci...
research
07/19/2021

### Perfectly Matched Sets in Graphs: Hardness, Kernelization Lower Bound, and FPT and Exact Algorithms

In an undirected graph G=(V,E), we say (A,B) is a pair of perfectly matc...