Enumeration Kernelizations of Polynomial Size for Cuts of Bounded Degree

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 ∈ A has at most d neighbors in B and every v ∈ 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 ≥ 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.