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A separator theorem for nonplanar graphs. (English) Zbl 0747.05051
Let $$G$$ be an $$n$$-vertex graph with no minor, a graph which can be obtained from a subgraph of $$G$$ by contracting edges, isomorphic to an $$h$$-vertex complete graph. The authors prove that the vertices of $$G$$ can be partitioned into three sets $$A$$, $$B$$, $$C$$ such that no edge joins a vertex in $$A$$ with a vertex in $$B$$, neither $$A$$ nor $$B$$ contains more than $$2n/3$$ vertices, and $$C$$ contains no more than $$h^{3/2}$$ $$n^{1/2}$$ vertices. This extends a theorem of R. J. Lipton and R. E. Tarjan [SIAM J. Appl. Math. 36, 177-189 (1979; Zbl 0432.05022)].
Reviewer: M.Hager (Leonberg)

##### MSC:
 05C40 Connectivity 05C10 Planar graphs; geometric and topological aspects of graph theory 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
separator theorem; nonplanar graphs; minor
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##### References:
 [1] John R. Gilbert, Joan P. Hutchinson, and Robert Endre Tarjan, A separator theorem for graphs of bounded genus, J. Algorithms 5 (1984), no. 3, 391 – 407. · Zbl 0556.05022 · doi:10.1016/0196-6774(84)90019-1 · doi.org [2] Richard J. Lipton and Robert Endre Tarjan, A separator theorem for planar graphs, SIAM J. Appl. Math. 36 (1979), no. 2, 177 – 189. · Zbl 0432.05022 · doi:10.1137/0136016 · doi.org [3] P. D. Seymour and Robin Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993), no. 1, 22 – 33. · Zbl 0795.05110 · doi:10.1006/jctb.1993.1027 · doi.org [4] N. Alon, P. D. Seymour, and R. Thomas, A separator theorem for graphs with an excluded minor and its applications (Proc. 22nd STOC, Baltimore, Maryland, 1990), ACM Press, 293-299.
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