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Unformatted text preview: Computational Topology (Jeff Erickson) Graph Separators In the spring of 1930,...König told me that he was about to finish a book that would include all that was known about graphs. I assured him that such a book would fill a great need; and I brought up my n-Arc Theorem which, having been published as a lemma in a curve-theoretical paper, had not yet come to his attention. König was greatly interested, but did not believe the theorem was correct. “This evening,” he said to me in parting, “I won’t go to sleep before having constructed a counterexample!”. When me met again the next day he greeted me with the words “Sleepless night!” — Karl Menger, “On the origin of the n-arc theorem”, J. Graph Theory 5:341–350, 1981. 10 Graph Separators “Divide and conquer” is one of the oldest and most widely used techniques for designing efficient algo- rithms. Divide-and-conquer algorithms partition their inputs into two or more independent subproblems, solve those subproblems recursively, and then combine the solutions to those subproblems to obtain their final output. This strategy can be successfully applied to several graph problems, provided we can quickly separate the graph into roughly equal subgraphs. An "-separator of an n-vertex graph G = ( V , E ) is a subset S ⊆ V such that each connected component of G \ S has at most " n vertices. Our goal is to find "-separators, for some constant 1 / 2 ≤ " < 1, that have few vertices. For example, any path has a 1 / 2-separator consisting of a single vertex; any binary tree has a 2 / 3-separator consisting of a single vertex; and any outer-planar graph has a 2 / 3-separator consisting of two vertices. The following classical theorem of Menger [ 12 ] , which is both a precursor and an easy consequence of the maxflow-mincut theorem, is a key tool in proving the existence of small separators. Theorem 10.1 (Menger). Let G = ( V , E ) be a graph. The minimum number of vertices separating any subsets A , b ⊆ V is equal to the maximum number of vertex-disjoint paths from A to B . 10.1 Planar Separators In the late 1970s, Richard Lipton and Robert Tarjan [ 11 ] proved the following seminal result. The Planar Separator Theorem. Any n-vertex planar graph has a 2 / 3-separator containing at most p 8 n vertices. Proof (Alon, Seymour, and Thomas [ 2 ] ): Let G be an embedded planar graph with n ≥ 3 vertices, and let k = b p 2 n c . Without loss of generality, we can assume that G has no loops or parallel edges, and that every face is a triangle bounded by three distinct edges. For any simple cycle C in G , let In ( C ) and Out ( C ) denote the vertices inside and outside C , respectively. No vertex of In ( C ) is adjacent to any vertex of Out ( C ) . Let C be a simple cycle satisfying three conditions: (1) C has at most 2 k vertices....
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This note was uploaded on 01/24/2012 for the course CS 598 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08