Faster parameterized algorithms for modification problems to minor-closed classes

Let G be a minor-closed graph class and let G be an n-vertex graph. We say that G is a k-apex of G if G contains a set S of at most k vertices such that G S belongs to G. Our first result is an algorithm that decides whether G is a k-apex of G in time 2^{ poly (k) } n², where poly is a polynomial function depending on G. This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos ICALP 2020, whose running time was 2^{ poly (k) } n³. The elimination distance of G to G, denoted by ed ₆ (G), is the minimum number of rounds required to reduce each connected component of G to a graph in G by removing one vertex from each connected component in each round. Bulian and Dawar Algorithmica 2017 provided an FPT-algorithm, with parameter k, to decide whether ed ₆ (G) k. However, its dependence on k is not explicit. We extend the techniques used in the first algorithm to decide whether ed ₆ (G) k in time 2^2^{2^{{ poly (k) }}} n². This is the first algorithm for this problem with an explicit parametric dependence in k. In the special case where G excludes some apex-graph as a minor, we give two alternative algorithms, running in time 2^2^{{ O (k² k) }} n² and 2^{ poly (k) } n³ respectively, where c and poly depend on G. As a stepping stone for these algorithms, we provide an algorithm that decides whether ed ₆ (G) k in time 2^{ O (tw k+ tw tw) } n, where tw is the treewidth of G. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs Eₖ (G) =\G ed ₆ (G) k\.