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A Separator Theorem for Planar Graphs
, 1977
"... Let G be any nvertex planar 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 2& & vertices. We exhibit an algorithm which finds su ..."
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Cited by 386 (1 self)
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Let G be any nvertex planar 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 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.
Dominating Sets in Planar Graphs
, 1994
"... Motivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs. Specifically, we wish to find the smallest ffl such that, for n sufficiently large, every nvertex planar graph cont ..."
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Motivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs. Specifically, we wish to find the smallest ffl such that, for n sufficiently large, every nvertex planar graph contains a dominating set of size at most ffln. We prove that 1 4 ffl 1 3 , and we conjecture that ffl = 1 4 . For triangulated discs we obtain a tight bound of ffl = 1 3 . The upper bound proof yields a lineartime algorithm for finding an n/3  size dominating set.
Fast Programs for Finding Maximum Planar Subgraphs
"... A previously reported, parallel method for calculating approximately maximum planar subgraphs has shown promising results, but was slow. The results of an initial implementation on the Connection Machine 2 prompted an investigation of more efficient methods of implementing the procedure. An improved ..."
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A previously reported, parallel method for calculating approximately maximum planar subgraphs has shown promising results, but was slow. The results of an initial implementation on the Connection Machine 2 prompted an investigation of more efficient methods of implementing the procedure. An improved implementation uses many fewer processors for the calculation, with substantially shorter communication times. By running several copies of the basic calculation simultaneously, the program performs one hundred times faster than earlier implementations. 1 Introduction A planar graph is one that can be drawn in a plane with its edges intersecting only at the vertices. Given an arbitrary graph, the process of extracting the largest such planar graph and embedding it in a plane requires immense computational expense. Because of the NPhard nature of this problem [1], it is necessary to find a method that will locate approximately the largest planar subgraph with a reasonable amount of effort....