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A New Proof of the Flat Wall Theorem

We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r>0 be integers, and let R=49152t^{24}(40t^2+r). An r-wall is obtained from a (2r x r)-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no K_t minor, and let W be an R-wall in G. We prove that there exist a subset A of V(G) of size at most 12288t^{24} and an r-subwall W' of W such that V(W') is disjoint from A and W' is a flat wall in G-A in the following sense. There exists a separation (X,Y) of G-A such that X\cap Y is a subset of the vertex set of the cycle C' that bounds the outer face of W', V(W') is a subset of Y, every peg of W' belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X\cap Y drawn on the boundary of the disk in the order determined by C'. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X\cap Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks.