Enumeration Kernels of Polynomial Size for Cuts of Bounded Degree

TL;DR

This paper develops polynomial-size enumeration kernels for the d-CUT problem and its variants, using structural parameters like vertex cover, neighborhood diversity, and clique partition number, advancing enumeration kernelization techniques.

Contribution

It introduces new polynomial-delay enumeration kernels for d-CUT variants based on structural graph parameters, including bijective kernels for clique partition number.

Findings

Polynomial-delay enumeration kernels for ENUM d-CUT and ENUM MAX-d-CUT with vertex cover and neighborhood diversity.

Fully-polynomial enumeration kernels for ENUM MIN-d-CUT with vertex cover and neighborhood diversity.

Bijective enumeration kernels for all variants when parameterized by clique partition number.

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.