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.

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