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 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 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.