Harmless Sets in Sparse Classes

In the classic Target Set Selection problem, we are asked to minimise the number of nodes to activate so that, after the application of a certain propagation process, all nodes of the graph are active. Bazgan and Chopin [Discrete Optimization, 14:170–182, 2014] introduced the opposite problem, named Harmless Set, in which they ask to maximise the number of nodes to activate such that not a single additional node is activated. In this paper we investigate how sparsity impacts the tractability of Harmless Set. Specifically, we answer two open questions posed by the aforementioned authors, namely a) whether the problem is FPT on planar graphs and b) whether it is FPT parametrised by treewidth. The first question can be answered in the positive using existing meta-theorems on sparse classes, and we further show that Harmless Set not only admits a polynomial kernel, but that it can be solved in subexponential time. We then answer the second question in the negative by showing that the problem is W[1]-hard when parametrised by a parameter that upper bounds treewidth.