Parameterized lower bound and NP-completeness of some $H$-free Edge Deletion problems

TL;DR

This paper establishes NP-completeness and parameterized complexity lower bounds for the $H$-free Edge Deletion problem, showing it remains hard for certain classes of graphs $H$, thus advancing understanding of its computational complexity.

Contribution

The paper proves NP-completeness for $H$-free Edge Deletion when $H$ contains a component that is a tree or regular graph, and shows these problems lack subexponential algorithms under ETH.

Findings

NP-complete for graphs $H$ with specific components

No subexponential parameterized algorithms under ETH

Extends complexity understanding of $H$-free Edge Deletion

Abstract

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. We prove that $H$-free Edge Deletion is NP-complete if $H$ is a graph with at least two edges and $H$ has a component with maximum number of vertices which is a tree or a regular graph. Furthermore, we obtain that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$, unless Exponential Time Hypothesis fails.