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（Enumeration Kernelization）最早由Creignou等人[TOCS 2017]提出，后由Golovach等人[JCSS 2022]改进为两种不同变体：完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)，若割F=(A, B)满足：对任意u∈A，其在B中的邻居数不超过d；且对任意v∈B，其在A中的邻居数不超过d，则称F为G的度-d-割。判断图中是否存在度-d-割是经典的NP难问题，并在参数复杂性[Algorithmica 2021, IWOCA 2021]中已得到充分研究。该问题还推广了已被广泛研究的MATCHING CUT问题（令d=1），后者是多项式延迟枚举核化领域的核心问题。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（枚举所有极小d-割）与ENUM MAX-DEGREE-d-CUT（枚举所有极大度-d-割）。针对输入图的各种结构参数，我们为每个固定的d≥1证明：ENUM DEGREE-d-CUT与ENUM MAX-DEGREE-d-CUT存在多项式规模的多项式延迟枚举核化，ENUM MIN-DEGREE-d-CUT存在多项式规模的完全多项式枚举核。