A Polynomial Kernel for Line Graph Deletion

TL;DR

This paper presents a polynomial kernel of size O(k^5) for the Line-Graph-Edge Deletion problem, enabling efficient preprocessing to determine if a graph can be made a line graph by deleting at most k edges.

Contribution

It provides the first polynomial kernel for the problem, resolving an open question from 2013 about kernelization bounds for line graph editing.

Findings

Polynomial kernel with O(k^5) vertices for Line-Graph-Edge Deletion.

Answers an open problem from 2013 on kernelization.

Advances understanding of graph editing problems.

Abstract

The line graph of a graph $G$ is the graph $L(G)$ whose vertex set is the edge set of $G$ and there is an edge between $e,f\in E(G)$ if $e$ and $f$ share an endpoint in $G$. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most $k$ edges from the input graph $G$ such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with $\mathcal{O}(k^{5})$ vertices. This answers an open question posed by Falk H\"{u}ffner at Workshop on Kernels (WorKer) in 2013.