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 ${ \cal E}_k({ \cal G})=\{ G\mid{ \sf ed}_{ \cal G}(G)\leq k\}$.
Categories: cs.DS cs.CC math.CO