Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017] and was later refined by Golovach et al. [JCSS 2022] into two different variants: fully-polynomial enumeration kernelization and polynomial-delay enumeration kernelization. In this paper, we consider the DEGREE-d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = (A, B) is a degree-d-cut of G if every $u \in A$ has at most d neighbors in B and every $v \in B$ has at most d neighbors in A. Checking the existence of a degree-d-cut in a graph is a well-known NP-hard problem and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA 2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d = 1) that has been a central problem in the literature of polynomial-delay enumeration kernelization. In this paper, we study three different enumeration variants of this problem, ENUM DEGREE-d-CUT, ENUM MIN-DEGREE-d-CUT and ENUM MAX-DEGREE-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal degree-d-cuts respectively. We consider various structural parameters of the input and for every fixed $d \geq 1$, we provide polynomial-delay enumeration kernelizations of polynomial size for ENUM DEGREE-d-CUT and ENUM MAX-DEGREE-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-DEGREE-d-CUT.

翻译：枚举核化最早由Creignou等人[TOCS 2017]提出，后由Golovach等人[JCSS 2022]细化为两种不同变体：全多项式枚举核化与多项式延迟枚举核化。本文从（多项式延迟）枚举核化的角度研究DEGREE-d-CUT问题。给定无向图G=(V, E)，割集F=(A, B)称为G的d-度割，当且仅当每个$u \in A$在B中至多有d个邻居，且每个$v \in B$在A中至多有d个邻居。检查图中是否存在d-度割是著名的NP难问题，并在参数化复杂度领域得到充分研究[Algorithmica 2021, IWOCA 2021]。该问题同时推广了经典问题MATCHING CUT（令d=1），后者一直是多项式延迟枚举核化研究中的核心问题。本文研究该问题的三种不同枚举变体：ENUM DEGREE-d-CUT（枚举所有d-割）、ENUM MIN-DEGREE-d-CUT（枚举所有最小d-割）以及ENUM MAX-DEGREE-d-CUT（枚举所有最大d-割）。我们考虑输入图的多种结构参数，针对每个固定$d \geq 1$，为ENUM DEGREE-d-CUT和ENUM MAX-DEGREE-d-CUT提供多项式大小的多项式延迟枚举核化，并为ENUM MIN-DEGREE-d-CUT提供多项式大小的全多项式枚举核。