The Harmless Set Problem

TL;DR

This paper investigates the computational complexity of the Harmless Set problem across various graph classes and parameters, providing new fixed-parameter algorithms and hardness results that deepen understanding of its tractability.

Contribution

The paper offers the first fixed-parameter algorithms for Harmless Set with respect to neighborhood diversity, twin cover, and vertex integrity, and establishes W[1]-hardness for several structural parameters.

Findings

W[1]-hard parameterized by feedback vertex set, pathwidth, treedepth, and cluster vertex deletion.

Polynomial-time solvable on graphs with bounded cliquewidth.

Fixed-parameter tractable on planar graphs for solution size.

Abstract

Given a graph $G = (V,E)$, a threshold function $t~ :~ V \rightarrow \mathbb{N}$ and an integer $k$, we study the Harmless Set problem, where the goal is to find a subset of vertices $S \subseteq V$ of size at least $k$ such that every vertex $v\in V$ has less than $t(v)$ neighbors in $S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity. Our focus lies on parameters that measure the structural properties of the input instance. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number. We also show that the Harmless Set problem with majority thresholds is W[1]-hard when parameterized by the treewidth of the input graph. We prove that the Harmless Set problem can be solved in polynomial time on graph with bounded cliquewidth. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to neighbourhood diversity, twin cover and vertex integrity of the input graph. We show that the problem parameterized by the solution size is fixed parameter tractable on planar graphs. We thereby resolve two open questions stated in C. Bazgan and M. Chopin (2014) concerning the complexity of {\sc Harmless Set} parameterized by the treewidth of the input graph and on planar graphs with respect to the solution size.