# Edge Deletion to Restrict the Size of an Epidemic

Given a graph G=(V,E), a set ℱ of forbidden subgraphs, we study ℱ-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any F∈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 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 ℱ is the set 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 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.

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