# On Kernelization for Edge Dominating Set under Structural Parameters

## TL;DR

This paper investigates the possibility of polynomial kernelizations for the Edge Dominating Set problem under various structural parameters, revealing both impossibility results and a complete classification for certain graph decompositions.

## Contribution

It provides a full classification of when polynomial kernels exist for EDS parameterized by the deletion distance to graph classes, including surprising positive results for certain path-based parameters.

## Key findings

- Polynomial kernels do not exist for deletion distance to disjoint unions of P3.
- A polynomial kernel exists for deletion distance to disjoint unions of P5.
- Complete classification for kernelizability based on graph class parameters.

## Abstract

In the NP-hard Edge Dominating Set problem (EDS) we are given a graph $G=(V,E)$ and an integer $k$, and need to determine whether there is a set $F\subseteq E$ of at most $k$ edges that are incident with all (other) edges of $G$. It is known that this problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by $k$. A caveat for this parameter is that it needs to be large, i.e., at least equal to half the size of a maximum matching of $G$, for instances not to be trivially negative. Motivated by this, we study the existence of polynomial kernels for EDS when parameterized by structural parameters that may be much smaller than $k$. Unfortunately, at first glance this looks rather hopeless: Even when parameterized by the deletion distance to a disjoint union of paths $P_3$ of length two there is no polynomial kernelization (under standard assumptions), ruling out polynomial kernels for many smaller parameters like the feedback vertex set size. In contrast, somewhat surprisingly, there is a polynomial kernelization for deletion distance to a disjoint union of paths $P_5$ of length four. As our main result, we fully classify for all finite sets $\mathcal{H}$ of graphs, whether a kernel size polynomial in $|X|$ is possible when given $X$ such that each connected component of $G-X$ is isomorphic to a graph in $\mathcal{H}$.