Smaller kernels for 3-leaf power modifications problems

Abstract

A graph G is a 3-leaf power if there is a tree T and a bijective mapping f from $V\left(G\right)$ to the set of leaves of T such that $(u,v)\in E\left(G\right)$ if and only if the distance in T between $f\left(u\right)$ and $f\left(v\right)$ is at most 3 for every distinct $u,v\in V\left(G\right)$. In the 3-Leaf Power Vertex Deletion (resp., 3-Leaf Power Completion) problem the input is a graph G and an integer k, and the goal is to decide whether G can be transformed into a 3-leaf power graph by deleting at most k vertices (resp., adding at most k edges). In this paper we give a kernel for 3-Leaf Power Vertex Deletion with $O\left(k^6\right)$ vertices and a kernel for 3-Leaf Power Completion with $O\left(k^2\right)$ vertices. Our results improve the previous $O\left(k^{14}\right)$-vertices kernel for 3-Leaf Power Vertex Deletion [Ahn et al., 2023 [3]] and the $O\left(k^3\right)$-vertices kernel for 3-Leaf Power Completion [Bessy et al., 2010 [5]].