Given a family $\mathcal{F}$ of graphs, a graph is \emph{$\mathcal{F}$-subgraph-free} if it has no subgraph isomorphic to a member of $\mathcal{F}$. We present a fixed-parameter linear-time algorithm that decides whether a planar graph can be made $\mathcal{F}$-subgraph-free by deleting at most $k$ vertices or $k$ edges, where the parameters are $k$, $\lvert \mathcal{F} \rvert$, and the maximum number of vertices in a member of $\mathcal{F}$. The running time of our algorithm is double-exponential in the parameters, which is faster than the algorithm obtained by applying the first-order model checking result for graphs of bounded twin-width. To obtain this result, we develop a unified framework for designing algorithms for this problem on graphs with a ``product structure.'' Using this framework, we also design algorithms for other graph classes that generalize planar graphs. Specifically, the problem admits a fixed-parameter linear time algorithm on disk graphs of bounded local radius, and a fixed-parameter almost-linear time algorithm on graphs of bounded genus. Finally, we show that our result gives a tight fixed-parameter algorithm in the following sense: Even when $\mathcal{F}$ consists of a single graph $F$ and the input is restricted to planar graphs, it is unlikely to drop any parameters $k$ and $\lvert V(F) \rvert$ while preserving fixed-parameter tractability, unless the Exponential-Time Hypothesis fails.

翻译：给定图族$\mathcal{F}$，若一个图不包含任何与$\mathcal{F}$中成员同构的子图，则称该图为\emph{$\mathcal{F}$-子图无关}图。本文提出一种固定参数线性时间算法，用于判定平面图是否可通过删除至多$k$个顶点或$k$条边成为$\mathcal{F}$-子图无关图，其中参数包括$k$、$\lvert \mathcal{F} \rvert$以及$\mathcal{F}$中成员的最大顶点数。该算法的运行时间关于参数呈双指数级，优于通过有界孪生宽度图的一阶模型检测结果所导出的算法。为获得此结果，我们开发了一个统一框架，用于在具有“乘积结构”的图上设计该问题的算法。基于此框架，我们还为其他推广平面图的图类设计了算法。具体而言，该问题在有界局部半径的圆盘图上存在固定参数线性时间算法，在有界亏格图上存在固定参数近似线性时间算法。最后，我们证明该结果提供了紧致的固定参数算法：即使当$\mathcal{F}$仅包含单个图$F$且输入限制为平面图时，若要在保持固定参数可解性的同时省略参数$k$或$\lvert V(F) \rvert$，除非指数时间假设不成立，否则几乎不可能实现。