On the (non-)existence of polynomial kernels for Pl-free edge modification problems

TL;DR

This paper investigates the existence of polynomial kernels for edge modification problems related to P_l-free graphs, providing both positive results for cographs and negative results for larger path and cycle-free classes.

Contribution

It establishes cubic vertex kernels for cograph edge modification problems and shows polynomial kernels are unlikely for P_l-free and Cl-free edge-deletion problems.

Findings

Cubic vertex kernels for cograph edge modification problems.

Polynomial kernels unlikely for P_l-free and Cl-free edge-deletion problems.

Addresses open questions on kernelization for specific graph classes.

Abstract

Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E \vartriangle F) satisfies the property P. In the P edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free and Cl-free edge-deletion problems for large enough l.