Graph modification of bounded size to minor-closed classes as fast as vertex deletion

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.