  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.
