Hitting minors on bounded treewidth graphs. III. Lower bounds

For a finite fixed collection of graphs the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists S subset of V (G) with vertical bar S vertical bar <= k such that G \ S does not contain any of the graphs in F as a minor. We provide lower bounds under the ETH on the smallest function f(F) such that F-M -DELETION can be solved in time f(F)(tw) . n(O(1)) on n-vertex graphs, where tw denotes the treewidth of G. We first prove that for any F containing connected graphs of size at least two, f(F)(tw) = 2(Omega(tw)), even if G is planar. Our main result is that if F contains a single connected graph H that is either P-5 tw or is not a minor of the banner, then f(F)(tw) = 2(Omega(tw.log tw)). (C) 2019 Elsevier Inc. All rights reserved.