A polynomial kernel for $3$-leaf power deletion

TL;DR

This paper presents a polynomial kernelization algorithm for the problem of deleting vertices to transform a graph into a 3-leaf power, reducing the problem size to polynomial bounds in parameter k.

Contribution

It introduces the first polynomial kernel for the 3-leaf power deletion problem, providing an efficient preprocessing step with size bounds polynomial in k.

Findings

Kernel size bounded by O(k^{14}) vertices.

Polynomial-time algorithm for kernelization.

Reduces problem complexity significantly.

Abstract

For a non-negative integer $\ell$, the $\ell$-leaf power of a tree $T$ is a simple graph $G$ on the leaves of $T$ such that two vertices are adjacent in $G$ if and only if their distance in $T$ is at most $\ell$. We provide a polynomial kernel for the problem of deciding whether we can delete at most $k$ vertices to make an input graph a $3$-leaf power of some tree. More specifically, we present a polynomial-time algorithm for an input instance $(G,k)$ for the problem to output an equivalent instance $(G',k')$ such that $k'\leq k$ and $G'$ has at most $O(k^{14})$ vertices.