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 ${\cal E}_k({\cal G})=\{G\mid{\sf ed}_{\cal G}(G)\leq k\}$.

翻译：令${\cal G}$为禁小图类，$G$为$n$个顶点的图。若$G$包含至多$k$个顶点的集合$S$使得$G\setminus S$属于${\cal G}$，则称$G$为${\cal G}$的$k$-apex。我们的第一个算法在时间$2^{{\sf poly}(k)}\cdot n^2$内判定$G$是否为${\cal G}$的$k$-apex，其中${\sf poly}$为依赖${\cal G}$的多项式函数。该算法改进了Sau、Stamoulis与Thilikos[ICALP 2020]此前运行时间为$2^{{\sf poly}(k)}\cdot n^3$的算法。$G$到${\cal G}$的消去距离${\sf ed}_{\cal G}(G)$定义为：每轮从每个连通分量删除一个顶点，将$G$的每个连通分量归约为${\cal G}$中图所需的最少轮数。Bulian与Dawar[Algorithmica 2017]给出了参数为$k$的FPT算法以判定${\sf ed}_{\cal G}(G)\leq k$，但其对$k$的依赖关系未显式给出。我们扩展首个算法的技术，在时间$2^{2^{2^{{\sf poly}(k)}}}\cdot n^2$内判定${\sf ed}_{\cal G}(G)\leq k$。这是该问题首个具有显式参数依赖性的算法。当${\cal G}$排除某些apex-图作为禁小图时，我们给出两个替代算法，运行时间分别为$2^{2^{{\cal O}(k^2\log k)}}\cdot n^2$和$2^{{\sf poly}(k)}\cdot n^3$，其中$c$与${\sf poly}$依赖${\cal G}$。作为这些算法的铺垫，我们给出一个算法在时间$2^{{\cal O}({\sf tw}\cdot k+{\sf tw}\log{\sf tw})}\cdot n$内判定${\sf ed}_{\cal G}(G)\leq k$，其中${\sf tw}$为$G$的树宽。最后，我们给出图类${\cal E}_k({\cal G})=\{G\mid{\sf ed}_{\cal G}(G)\leq k\}$的禁小图集合中图尺寸的显式上界。