Polynomial-Size Enumeration Kernelizations for Long Path Enumeration

TL;DR

This paper introduces polynomial-size enumeration kernels for the ENUM LONG-PATH problem, leveraging structural graph parameters to efficiently enumerate all paths of a given length.

Contribution

It provides the first polynomial-delay enumeration kernels for ENUM LONG-PATH based on vertex cover, dissociation number, and distance to clique parameters.

Findings

Polynomial-delay enumeration kernels of polynomial size for vertex cover.

Extension of enumeration kernelization to dissociation number and distance to clique.

Improved efficiency in enumerating all long paths in graphs.

Abstract

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.