Polynomial kernels for 3-leaf power graph modification problems

Abstract

A graph G = ( V , E ) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that u v ∈ E 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 FPT when parameterized by the size of the edge set modification. However, polynomial kernels were known for none of these three problems. For each of them, we provide kernels with O ( k 3 ) vertices that can be computed in linear time. We thereby answer an open problem first mentioned by Dom et al. (2004) [8].