We introduce a series of graph decompositions based on the modulator/target scheme of modification problems that enable several algorithmic applications that parametrically extend the algorithmic potential of planarity. In the core of our approach is a polynomial time algorithm for computing planar H-modulators. Given a graph class H, a planar H-modulator of a graph G is a set X \subseteq V(G) such that the ``torso'' of X is planar and all connected components of G - X belong to H. Here, the torso of X is obtained from G[X] if, for every connected component of G-X, we form a clique out of its neighborhood on G[X]. We introduce H-Planarity as the problem of deciding whether a graph G has a planar H-modulator. We prove that, if H is hereditary, CMSO-definable, and decidable in polynomial time, then H-Planarity is solvable in polynomial time. Further, we introduce two parametric extensions of H-Planarity by defining the notions of H-planar treedepth and H-planar treewidth, which generalize the concepts of elimination distance and tree decompositions to the class H. Combining this result with existing FPT algorithms for various H-modulator problems, we thereby obtain FPT algorithms parameterized by H-planar treedepth and H-planar treewidth for numerous graph classes H. By combining the well-known algorithmic properties of planar graphs and graphs of bounded treewidth, our methods for computing H-planar treedepth and H-planar treewidth lead to a variety of algorithmic applications. For instance, once we know that a given graph has bounded H-planar treedepth or bounded H-planar treewidth, we can derive additive approximation algorithms for graph coloring and polynomial-time algorithms for counting (weighted) perfect matchings. Furthermore, we design Efficient Polynomial-Time Approximation Schemes (EPTAS-es) for several problems, including Maximum Independent Set.

翻译：我们基于修改问题的调节器/目标方案引入了一系列图分解方法，这些方法支持多种算法应用，从而参数化地扩展了平面性的算法潜力。我们方法的核心是计算平面H-调节器的多项式时间算法。给定图类H，图G的平面H-调节器是一个集合X ⊆ V(G)，使得X的“躯干”是平面的，且G - X的所有连通分支都属于H。此处，X的躯干由G[X]通过以下方式得到：对于G-X的每个连通分支，我们在G[X]中将其邻域形成一个团。我们引入H-平面性问题，即判定图G是否具有平面H-调节器。我们证明，若H是遗传的、可由CMSO公式定义且可在多项式时间内判定，则H-平面性可在多项式时间内求解。进一步，我们通过定义H-平面树深和H-平面树宽的概念，引入了H-平面性的两个参数化扩展，将消除距离和树分解的概念推广到图类H。结合这一结果与针对各类H-调节器问题的现有FPT算法，我们由此获得了以H-平面树深和H-平面树宽为参数的FPT算法，适用于众多图类H。通过结合平面图和有界树宽图的已知算法性质，我们计算H-平面树深和H-平面树宽的方法带来了多种算法应用。例如，一旦已知给定图具有有界H-平面树深或有界H-平面树宽，我们可以推导出图着色的加法近似算法以及计算（加权）完美匹配的多项式时间算法。此外，我们为包括最大独立集在内的多个问题设计了高效多项式时间近似方案（EPTAS）。