Results 1  10
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18
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
Abstract

Cited by 27 (10 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
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 (6 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...
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 ..."
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Cited by 23 (7 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].
Checking the Convexity of Polytopes and the Planarity of Subdivisions
, 1998
"... This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivi ..."
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Cited by 23 (5 self)
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This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required
Recognizing String Graphs is Decidable
"... A graph is called a string graph if its vertices can be represented by continuous curves ("strings") in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n) with the property that every string g ..."
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Cited by 13 (2 self)
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A graph is called a string graph if its vertices can be represented by continuous curves ("strings") in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n) with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n) times. We obtain as a corollary...
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 11 (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
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
Planar Graphs with Topological Constraints
 Journal of Graph Algorithms and Applications
, 2002
"... We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, userdefined topological constraints. The constraints consist each of a cycle... ..."
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Cited by 5 (0 self)
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We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, userdefined topological constraints. The constraints consist each of a cycle...
SPLITTING NUMBER is NPcomplete
, 1997
"... We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such ope ..."
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Cited by 4 (4 self)
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We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v 1 and v 2 , and attaches the neighbors of v either to v 1 or to v 2 . We prove that the splitting number decision problem is NPcomplete. We obtain as a consequence that planar subgraph remains NPcomplete when restricted to graphs with maximum degree 3, when restricted to graphs with no subdivision of K 5 , or when restricted to cubic graphs, problems that have been open since 1979.
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
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Cited by 3 (0 self)
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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...