Hitting minors on bounded treewidth graphs. I. General upper bounds

For a finite collection of graphs F, the F-M-Deletion problem consists in, given a graph G and an integer k, deciding 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 f_F such that F-M-Deletion can be solved in time f_F(tw) · n^O(1) on n-vertex graphs. We prove that f_F(tw) = 2^2O(tw·log tw) for every collection F, that f_F(tw) = 2^O(tw·log tw) if F contains a planar graph, and that f_F(tw) = 2^O(tw) if in addition the input graph G is planar or embedded in a surface. We also consider the version of the problem where the graphs in F are forbidden as topological minors, called F-TM-Deletion. We prove similar results for this problem, except that in the last two algorithms, instead of requiring F to contain a planar graph, we need it to contain a subcubic planar graph. This is the first of a series of articles on this topic.