Let $\Pi$ be a hereditary graph class. The problem of deletion to $\Pi$, takes as input a graph $G$ and asks for a minimum number (or a fixed integer $k$) of vertices to be deleted from $G$ so that the resulting graph belongs to $\Pi$. This is a well-studied problem in paradigms including approximation and parameterized complexity. Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether $k$ vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be FPT as long as the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. While this was shown using some black box theorems, faster algorithms were shown when each of the hereditary graph classes has a finite forbidden set. In this paper, we do a deep dive on pairs of specific graph classes ($\Pi_1, \Pi_2$) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. We design a general FPT algorithm and approximation algorithm for pairs of graph classes (possibly having infinite forbidden sets) satisfying certain conditions. These algorithms cover several pairs of popular graph classes. Our algorithm makes non-trivial use of the branching technique and as a black box, FPT algorithms for deletion to individual graph classes.

翻译：令 $\Pi$ 为一遗传图类。删去至 $\Pi$ 问题以图 $G$ 为输入，要求删除最少数量的顶点（或给定整数 $k$ 个顶点），使得结果图属于 $\Pi$。这是一个在近似算法和参数化复杂性等范式中被广泛研究的问题。近期，该问题的一个自然扩展被提出：给定一组有限遗传图类，目标判定能否从给定图中删除 $k$ 个顶点，使得结果图的每个连通分量属于某一给定遗传图类。已证明该问题是FPT的，前提是每个给定遗传图类的删去问题均为固定参数可解的，且属于任一图类的性质可在计数单子二阶逻辑（CMSO）中表达。尽管这一结论通过若干黑箱定理得到，但当每个遗传图类具有有限禁止集时，可设计更快算法。本文深入研究特定图类对 ($\Pi_1, \Pi_2$) 的情况，要求结果图连通分量分属这些图类，并设计更简单高效的FPT算法。我们为满足特定条件的图类对（可能具有无限禁止集）设计了一个通用FPT算法与近似算法。这些算法覆盖了多组常见图类对。我们的算法非平凡地运用了分支技术，并将删去至单个图类的FPT算法作为黑箱使用。