Results 1  10
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30
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
Abstract

Cited by 33 (13 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
Abstract

Cited by 31 (8 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
On Winning Strategies With Unary Quantifiers
 J. Logic and Computation
, 1996
"... A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in firstorder logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q ..."
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Cited by 25 (5 self)
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A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in firstorder logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q u is the set of all unary generalized quantifiers. Using this combinatorial argument and a combination of secondorder EhrenfeuchtFra iss'e games developed by Ajtai and Fagin, we prove that connectivity of finite structures is not in monadic \Sigma 1 1 with any set of unary quantifiers, even if sentences are allowed to contain builtin relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of finite graphs defined by forbidden minors, is in L !! (Q u ). Especially, the class of planar graphs is not in L !! (Q u ). 1. Introduction The expressive power of firstorder logic L !! is rather limited. This is beca...
Parameteried complexity of constraint satisfaction problems
 In Proceedings of the 19th Annual IEEE Conference on Computational Complexity
"... We prove a parameterized analog of Schaefer’s Dichotomy Theorem: we show that for every finite boolean constraint family F, deciding whether a formula containing constraints from F has a satisfying assignment of weight exactly k is either fixedparameter tractable (FPT) or W[1]complete. We give a si ..."
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Cited by 21 (2 self)
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We prove a parameterized analog of Schaefer’s Dichotomy Theorem: we show that for every finite boolean constraint family F, deciding whether a formula containing constraints from F has a satisfying assignment of weight exactly k is either fixedparameter tractable (FPT) or W[1]complete. We give a simple characterization of those constraints that make the problem fixedparameter tractable. The special cases when the formula is restricted to be bounded occurrence, bounded treewidth or planar are also considered, it turns out that in these cases the problem is in FPT for every constraint family F. 1
A simple lineartime modular decomposition algorithm for graphs, using order extension
, 2004
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Simultaneous embedding of a planar graph and its dual on the grid
, 2002
"... Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar ..."
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Cited by 19 (10 self)
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Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straightline segments and the only crossings are between primaldual pairs of edges. We provide a lineartime algorithm that simultaneously embeds a 3connected planar graph and its dual on a (2n −2) ×(2n −2) integer grid, where n is the total number of vertices in the graph and its dual. Furthermore our embedding algorithm satisfies the two natural requirements mentioned above.
L.: Finding a maximumgenus graph imbedding
 Journal of the ACM
, 1988
"... The complltatioual complexity of ("oBst.ructing the imbeddings of a given graph int.o surfaccs of different genus is not wellunderstood. In t.his paper, topological method:! and a reduction to linear matroid parity are u~cd to develop a polynoIllialtime algorithm to find a ma.ximumgenus cell ..."
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Cited by 14 (1 self)
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The complltatioual complexity of ("oBst.ructing the imbeddings of a given graph int.o surfaccs of different genus is not wellunderstood. In t.his paper, topological method:! and a reduction to linear matroid parity are u~cd to develop a polynoIllialtime algorithm to find a ma.ximumgenus cellular imbedding. This seems to be the fin;t imbedding algorithm fot ' which the mUlling time is not expollential in the genus of the imbctlding surface. 1.
Graph Minors and Graphs on Surfaces
, 2001
"... Graph minors and the theory of graphs embedded in surfaces are ..."
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Cited by 8 (3 self)
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Graph minors and the theory of graphs embedded in surfaces are