Optimal Algorithms for Hitting (Topological) Minors on Graphs of Bounded Treewidth

For a fixed collection of graphs F , the F-M-Deletion problem consists in, given a graph G and an integer k, decide whether there exists S ⊆ V (G) with |S| ≤ k such that G \ S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-Deletion when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F , the smallest function fF such that F-M-Deletion can be solved in time fF(tw) ·n on n-vertex graphs. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in F are connected and at least one of them is planar, then fF(w) = 2 . When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds: • f{K4}(w) = 2 . • f{Ci}(w) = 2 Θ(w) for every i ≤ 4, and f{Ci}(w) = 2 Θ(w·logw) for every i ≥ 5. • f{Pi}(w) = 2 Θ(w) for every i ≤ 4, and f{Pi}(w) = 2 Θ(w·logw) for every i ≥ 6. The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove that essentially the same set of results holds.