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.

翻译：鉴于一个GG = (V,E) 美元, 一个阈值函数 $t~ : ~ V\ rightrow \ mathbb{N} 美元和一个整数美元, 我们研究“ 无伤害的Set ” 问题, 目标是找到一个脊椎的子集 $S = subseteq V$, 其大小至少为 $k$, 使每个顶点的美元值均低于 $(v) 邻居 $S 。 我们从参数化的复杂度角度出发, 提高我们对问题的理解。 我们的焦点在于测量输入实例的结构属性的参数 。 我们显示, 问题是 W[1] 硬性参数, 由一系列相当限制性的结构参数组成, 如反馈的顶点数、 路边、 树深度, 甚至最小的顶点的顶点是 。 我们的边端图显示W[1] 硬性数值, 和 硬性的直径比值是 硬性 。