Edge Deletion to Restrict the Size of an Epidemic

Given a graph $G=(V,E)$, a set $\mathcal{F}$ of forbidden subgraphs, we study $\mathcal{F}$-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any $F\in \mathcal{F}$ as a subgraph. For the parameter treewidth, the question of whether the problem is FPT has remained open. Here we give a negative answer by showing that the problem is W[1]-hard when parameterized by the treewidth, which rules out FPT algorithms under common assumption. Thus we give a solution to the conjecture posted by Jessica Enright and Kitty Meeks in [Algorithmica 80 (2018) 1857-1889]. We also prove that the $\mathcal{F}$-Free Edge Deletion problem is W[2]-hard when parameterized by the solution size $k$, feedback vertex set number or pathwidth of the input graph. A special case of particular interest is the situation in which $\mathcal{F}$ is the set $\mathcal{T}_{h+1}$ of all trees on $h+1$ vertices, so that we delete edges in order to obtain a graph in which every component contains at most $h$ vertices. This is desirable from the point of view of restricting the spread of disease in transmission network. We prove that the $\mathcal{T}_{h+1}$-Free Edge Deletion problem is fixed-parameter tractable (FPT) when parameterized by the vertex cover number. We also prove that it admits a kernel with $O(hk)$ vertices and $O(h^2k)$ edges, when parameterized by combined parameters $h$ and the solution size $k$.