A replacement action is a function $\mathcal{L}$ that maps each graph $H$ to a collection of graphs of size at most $|V(H)|$. Given a graph class $\mathcal{H}$, we consider a general family of graph modification problems, called $\mathcal{L}$-Replacement to $\mathcal{H}$, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in $\mathcal{L}(H_1)$ so that the resulting graph belongs to $\mathcal{H}$. $\mathcal{L}$-Replacement to $\mathcal{H}$ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves $\mathcal{L}$-Replacement to $\mathcal{H}$ in time $2^{{\rm poly}(k)}\cdot |V(G)|^2$ for every minor-closed graph class $\mathcal{H}$, where {\rm poly} is a polynomial whose degree depends on $\mathcal{H}$, under a mild technical condition on $\mathcal{L}$. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to $\mathcal{H}$ within the same running time. Our second algorithm is an improvement of the first one when $\mathcal{H}$ is the class of graphs embeddable in a surface of Euler genus at most $g$ and runs in time $2^{\mathcal{O}(k^{9})}\cdot |V(G)|^2$, where the $\mathcal{O}(\cdot)$ notation depends on $g$. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

翻译：替换操作是一个函数$\mathcal{L}$，它将每个图$H$映射至规模不超过$|V(H)|$的图集合。给定图类$\mathcal{H}$，我们考虑一类广义的图修改问题，称为$\mathcal{L}$-替换至$\mathcal{H}$，其输入为一个图$G$，问题在于是否可以通过将$G$中某个规模不超过$k$个顶点的诱导子图$H_1$替换为$\mathcal{L}(H_1)$中的图$H_2$，使得所得图属于$\mathcal{H}$。$\mathcal{L}$-替换至$\mathcal{H}$能够模拟多种图修改问题，包括顶点删除、边删除/添加/编辑/收缩、顶点标识、子图补全、独立集删除、（诱导）匹配删除/收缩等。我们提出两种算法。第一种算法在$\mathcal{L}$满足温和技术条件的前提下，对于任意子式封闭图类$\mathcal{H}$，可在$2^{{\rm poly}(k)}\cdot |V(G)|^2$时间内求解$\mathcal{L}$-替换至$\mathcal{H}$问题，其中{\rm poly}为次数依赖于$\mathcal{H}$的多项式。该结果推广了Morelle、Sau、Stamoulis和Thilikos [ICALP 2020, ICALP 2023]针对顶点删除至$\mathcal{H}$这一特例的结论，并保持了相同的时间复杂度。当$\mathcal{H}$为可嵌入欧拉亏格不超过$g$的曲面图类时，我们的第二种算法改进了第一种算法，其运行时间为$2^{\mathcal{O}(k^{9})}\cdot |V(G)|^2$，其中$\mathcal{O}(\cdot)$记号依赖于$g$。据我们所知，这是针对如此广义的图修改问题族至子式封闭类，首次提出的具有合理参数依赖性的参数化算法。