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A Separator Theorem for Nonplanar Graphs

Noga Alon, Paul Seymour and Robin Thomas
Journal of the American Mathematical Society
Vol. 3, No. 4 (Oct., 1990), pp. 801-808
DOI: 10.2307/1990903
Stable URL: http://www.jstor.org/stable/1990903
Page Count: 8
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A Separator Theorem for Nonplanar Graphs
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Abstract

Let G be an n-vertex graph with no minor isomorphic to an h-vertex complete graph. We 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 h3/2n1/2 vertices. This extends a theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds such a partition (A, B, C,) in time O(h1/2n1/2 m), where m = ∣ V(G)∣ + ∣ E(G)∣.

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