Results 1  10
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278
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 ..."
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Cited by 461 (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.
Upward Planarity Testing
 SIAM Journal on Computing
, 1995
"... Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upw ..."
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Cited by 87 (13 self)
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Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and singlesource digraphs. We also sketch the proof of NPcompleteness of upward planarity testing.
Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
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Cited by 78 (0 self)
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We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
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Cited by 75 (7 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth
, 2008
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Planar Orientations with Low OutDegree and Compaction of Adjacency Matrices
 Theoretical Computer Science
, 1991
"... We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounde ..."
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Cited by 46 (5 self)
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We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3bounded orientation, which can be constructed in linear time. ffl A 6bounded acyclic orientation, and a 3bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...
A Lineartime Algorithm for Drawing a Planar Graph on a Grid
 Information Processing Letters
, 1989
"... We present a lineartime algorithm that, given an nvertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straightline segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid i ..."
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Cited by 46 (4 self)
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We present a lineartime algorithm that, given an nvertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straightline segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid in the plane in such a way that the edges are straight, nonintersecting line segments. The existence of such straightline embeddings for planar graphs was independently discovered by F'ary [Fa48], Stein [St51], and Wagner [Wa36]; this result also follows from Steinitz's theorem on convex polytopes in three dimensions [SR34]. The first algorithms for constructing straightline embeddings [Tu63, CYN84, CON85] required highprecision arithmetic, and the resulting drawings were not very aesthetic, since they tend to produce uneven distributions of vertices over the drawing area. Rosenstiehl and Tarjan [RT86] noticed that it would be convenient to be able to map veritices of a planar graph into a...
The number of Reidemeister Moves Needed for Unknotting
, 2008
"... There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embe ..."
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Cited by 45 (11 self)
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There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1skeleton of the interior of a compact, orientable, triangulated PL 3manifold M. There is a positive constant c2 such that for each t ≥ 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.