Faster parameterized algorithms for modification problems to minor-closed classes

Let \({\cal G}\) be a minor-closed graph class and let \(G\) be an \(n\)-vertex graph. We say that \(G\) is a \(k\)-apex of \({\cal G}\) if \(G\) contains a set \(S\) of at most \(k\) vertices such that \(G\setminus S\) belongs to \({\cal G}\). Our first result is an algorithm that decides whether \(G\) is a \(k\)-apex of \({\cal G}\) in time \(2^{{\sf poly}(k)}\cdot n^2\), where \({\sf poly}\) is a polynomial function depending on \({\cal G}\). This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020], whose running time was \(2^{{\sf poly}(k)}\cdot n^3\). The elimination distance of \(G\) to \({\cal G}\), denoted by \({\sf ed}_{\cal G}(G)\), is the minimum number of rounds required to reduce each connected component of \(G\) to a graph in \({\cal G}\) 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 \({\sf ed}_{\cal G}(G)\leq k\). However, its dependence on \(k\) is not explicit. We extend the techniques used in the first algorithm to decide whether \({\sf ed}_{\cal G}(G)\leq k\) in time \(2^{2^{2^{{\sf poly}(k)}}}\cdot n^2\). This is the first algorithm for this problem with an explicit parametric dependence in \(k\). In the special case where \({\cal G}\) excludes some apex-graph as a minor, we give two alternative algorithms, running in time \(2^{2^{{\cal O}(k^2\log k)}}\cdot n^2\) and \(2^{{\sf poly}(k)}\cdot n^3\) respectively, where \(c\) and \({\sf poly}\) depend on \({\cal G}\). As a stepping stone for these algorithms, we provide an algorithm that decides whether \({\sf ed}_{\cal G}(G)\leq k\) in time \(2^{{\cal O}({\sf tw}\cdot k+{\sf tw}\log{\sf tw})}\cdot n\), where \({\sf tw}\) 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 \({\cal E}_k({\cal G})=\{G\mid{\sf ed}_{\cal G}(G)\leq k\}\).