Polynomial kernels for 3-leaf power graph modification problems

TL;DR

This paper introduces cubic polynomial kernels for 3-leaf power graph modification problems, providing the first known polynomial kernels for these problems and solving an open problem in the field.

Contribution

It presents the first polynomial kernels for edge modification problems on 3-leaf power graphs, specifically cubic kernels computable in linear time.

Findings

Cubic kernels for 3-leaf power edge modification problems.

Linear-time computability of these kernels.

Resolution of an open problem from 2005.

Abstract

A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u,v) is an edge iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, Huffner and Niedermeier (2005).