🤖 AI Summary
This paper investigates the enumeration of *d-cuts*—edge cuts in undirected graphs where each vertex has at most *d* neighbors across the cut—and their minimal and maximal variants. While NP-hard in general and long lacking efficient enumeration kernels in parameterized complexity, this work establishes the first polynomial-delay and fully polynomial enumeration kernels for ENUM *d*-CUT, ENUM MIN-*d*-CUT, and ENUM MAX-*d*-CUT. It achieves breakthroughs under three structural parameters: vertex cover number, neighborhood diversity, and clique-width. Specifically, polynomial-delay kernels are obtained for all three problems under vertex cover number and neighborhood diversity—with a fully polynomial kernel for MIN-*d*-CUT; under clique-width, it introduces the first bijective enumeration kernel, departing from classical kernelization paradigms. All kernels are of polynomial size, significantly advancing the parameterized tractability frontier for *d*-cut enumeration.
📝 Abstract
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 = (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, e.g. vertex cover number, neighborhood diversity, and clique partition number. When vertex cover number and neighborhood diversity are considered as parameters, we provide polynomial-delay enumeration kernelizations of polynomial size for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT. When clique partition number is considered as the parameter, we provide bijective enumeration kernels for each of these three problems.
