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Graph minors. XVI: Excluding a non-planar graph. (English) Zbl 1023.05040
Summary: This paper contains the cornerstone theorem of the series. We study the structure of graphs with no minor isomorphic to a fixed graph $$L$$, when $$L$$ is non-planar. (The case when $$L$$ is planar was studied in an earlier paper.) We find that every graph with no minor isomorphic to $$L$$ may be constructed by piecing together in a tree-structure graphs each of which “almost” embeds in some surface in which $$L$$ cannot be embedded.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C83 Graph minors
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##### References:
 [1] Dirac, G, A property of 4-chromatic graphs and remarks on critical graphs, J. London math. soc., 27, 85-92, (1952) · Zbl 0046.41001 [2] Duffin, R.J, Topology of series-parallel networks, J. math. anal. appl., 10, 303-318, (1965) · Zbl 0128.37002 [3] Erdös, P; Szekeres, G, A combinatorial problem in geometry, Composito. math., 2, 463-470, (1935) · Zbl 0012.27010 [4] Robertson, N; Seymour, P.D, Graph minors. V. excluding a planar graph, J. combin. theory ser. B, 41, 92-114, (1986) · Zbl 0598.05055 [5] Robertson, N; Seymour, P.D, Graph minors. IX. disjoint crossed paths, J. combin. theory ser. B, 49, 40-77, (1990) · Zbl 0741.05044 [6] Robertson, N; Seymour, P.D, Graph minors. X. obstructions to tree-decomposition, J. combin. theory ser. B, 52, 153-190, (1991) · Zbl 0764.05069 [7] Robertson, N; Seymour, P.D, Graph minors. XII. distance on a surface, J. combin. theory ser. B, 64, 240-272, (1995) · Zbl 0840.05016 [8] Robertson, N; Seymour, P.D, Graph minors. XIV. extending an embedding, J. combin. theory ser. B, 65, 23-50, (1995) · Zbl 0840.05017 [9] Robertson, N; Seymour, P.D, Graph minors. XV. giant steps, J. combin. theory ser. B, 68, 112-148, (1996) · Zbl 0860.05023 [10] Robertson, N; Seymour, P.D; Thomas, R, Quickly excluding a planar graph, J. combin. theory ser. B, 62, 323-348, (1994) · Zbl 0807.05023 [11] Seese, D.G; Wessel, W, Grids and their minors, J. combin. theory ser. B, 47, 349-360, (1989) · Zbl 0636.05023 [12] Wagner, K.A, Über eine eigenshaft der ebenen komplexe, Math. ann., 114, 570-590, (1937) · JFM 63.0550.01
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