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 d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = E(A, B) is a 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 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 d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal d-cuts respectively. We consider various structural parameters of the input and provide polynomial-delay enumeration kernels for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-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：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）枚举核化角度研究d-割问题。给定无向图G=(V,E)，割F=E(A,B)称为G的d-割，当且仅当A中每个顶点在B中至多有d个邻点，且B中每个顶点在A中至多有d个邻点。判定图中是否存在d-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]。该问题同时推广了已被深入研究的匹配割问题（设d=1），后者一直是多项式延迟枚举核化文献中的核心问题。本文研究了该问题的三种不同枚举变体：ENUM d-割、ENUM MIN-d-割与ENUM MAX-d-割，分别旨在枚举所有d-割、所有极小d-割以及所有极大d-割。我们针对输入图的多种结构参数进行分析，为ENUM d-割和ENUM MAX-d-割提供了多项式延迟枚举核，并为ENUM MIN-d-割提供了多项式大小的完全多项式枚举核。