Let ${\cal G}$ be a subclosed graph class and $G$ be a $n$-vertex graph. $G$ is $G\setminus S$ is ${ \calG}$. The first result is that $G$ is ${ Algorithm to determine if \cal G}$ is $k$-apex. {\sf poly}$ is a polynomial function that depends on ${\cal G}$. This algorithm improves on the previous algorithm given by Sau, Stamoulis, and Thilikos. [ICALP 2020], whose execution time was $2^{{\sf poly}(k)}\cdot n^3$ . The elimination distance from $G$ to ${\cal G}$, denoted by ${\sf ed}_{\cal G}(G)$, is the minimum is the number of rounds. To transform $ into a graph of ${\cal G}$, remove one vertex from each connected component in each round.Brian and Dawar [Algorithmica 2017]
${\sf ed}_{\cal G}(G)\leq k$. However, the dependency on $k$ is not explicit. ${\sf ed}_{\cal G}(G)\leq k$ is $2^{2^{2^{{\sf poly}(k) }}}\cdotn^2$. The first algorithm for this problem with explicit parametric dependencies on k$. In the special case where ${\cal G}$ excludes some vertex graphs as minors, we specify two alternative algorithms that run in time $2^{2^{{\cal O}(k ^2\log k)} $c$ and ${\sf poly}$ depend on ${\cal G}$. As a stepping stone for these algorithms, ${\sf ed}_{\cal G}(G)\leq k$ is assumed to be $2^{{\cal O}({\sf tw} \cdot k+{\sf tw} \log{\sf tw})}\cdot n$, where ${\sf tw}$ is the width of the tree in $G$. Finally, the graph class ${\cal E}_k({\cal G})=\{G\mid{\sf ed}_ {\cal G}(G)\leq k\}$.
