We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.

Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G \ S belongs to G. We denote by A k (G) the set of all graphs that are k-apices of G. We prove that every graph in the obstruction set of A k (G), i.e., the minor-minimal set of graphs not belonging to A k (G), has size at most 2 2 2 2 poly (k) , where poly is a polynomial function whose degree depends on the size of the minor-obstructions of G. This bound drops to 2 2 poly(k) when G excludes some apex graph as a minor.

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For a finite collection of graphs F, the F-TM-Deletion problem has as input an n-vertex graph G and an integer k and asks whether there exists a set S ⊆ V (G) with |S| ≤ k such that G \ S does not contain any of the graphs in F as a topological minor. We prove that for every such F, F-TM-Deletion is fixed parameter tractable on planar graphs. In particular, we provide an f (h, k) · n 2 algorithm where h is an upper bound to the vertices of the graphs in F.

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