Results 1  10
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50
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
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 18 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
Linkless embeddings of graphs in 3space
 BULLETIN OF THE AMER. MATH. SOC
, 1993
"... We announce results about flat (linkless) embeddings of graphs in 3space. A piecewiselinear embedding of a graph in 3space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of ..."
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Cited by 14 (2 self)
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We announce results about flat (linkless) embeddings of graphs in 3space. A piecewiselinear embedding of a graph in 3space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the complement in 3space of the embedding of every subgraph is free. (ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of K5 or K3,3. (iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by “3switches”, an analog of 2switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other. (iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from K6 by means of Y∆ and ∆Yexchanges.
Embedding and knotting of manifolds in Euclidean spaces
 London Math. Soc. Lect. Notes
"... Abstract. A clear understanding of topology of higherdimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higherdimensional topology in a way which makes clear the visual and algebraic constructions appear natu ..."
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Cited by 12 (7 self)
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Abstract. A clear understanding of topology of higherdimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higherdimensional topology in a way which makes clear the visual and algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities. More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to nonspecialists. We outline a general approach to embeddings via the classical van KampenShapiroWuHaefligerWeber ’deleted product ’ obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spie˙z and the author): a generalization the HaefligerWeber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range. 1.
Some recent progress and applications in graph minor theory
, 2006
"... In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are wellquasiordered under the graph minor relation. Recently, a n ..."
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Cited by 8 (4 self)
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In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are wellquasiordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed.
Embedding obstructions and 4dimensional thickenings of 2complexes
 Proc. Amer. Math. Soc
"... Abstract. The vanishing of Van Kampen’s obstruction is known to be necessary and sufficient for embeddability of a simplicial ncomplex into R 2n for n ̸ = 2, and it was recently shown to be incomplete for n = 2. We use algebraictopological invariants of fourmanifolds with boundary to introduce a ..."
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Cited by 5 (0 self)
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Abstract. The vanishing of Van Kampen’s obstruction is known to be necessary and sufficient for embeddability of a simplicial ncomplex into R 2n for n ̸ = 2, and it was recently shown to be incomplete for n = 2. We use algebraictopological invariants of fourmanifolds with boundary to introduce a sequence of higher embedding obstructions for a class of 2complexes in R 4. 1.
Bounded Combinatorial Width and Forbidden Substructures
, 1995
"... All rights reserved. This dissertation may not be reproduced in whole or in part, by mimeograph or other means, without the permission of the author. Supervisor: M. R. Fellows A substantial part of the history of graph theory deals with the study and classication of sets of graphs that share common ..."
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Cited by 4 (2 self)
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All rights reserved. This dissertation may not be reproduced in whole or in part, by mimeograph or other means, without the permission of the author. Supervisor: M. R. Fellows A substantial part of the history of graph theory deals with the study and classication of sets of graphs that share common properties. One predominant trend is to characterize graph families by sets of minimal forbidden graphs (within some partial ordering on the graphs). For example, the famous Kuratowski Theorem classi es the planar graph family by two forbidden graphs (in the topological partial order). Most, if not all, of the current approaches for nding these forbidden substructure characterizations use extensive and specialized case analysis. Thus, until now, for a xed graph family,thistype of mathematical theorem proving often required months or even years of human e ort. The main focus of this dissertation is to develop a practical theory for automating (with distributed computer programming) this classic part of graph theory. We extend and (more importantly) implement avariation
Intrinsically Triple Linked Complete Graphs
 Topology AppL
, 2001
"... We prove that every embedding of K 10 in R 3 contains a nonsplit link of three components. ..."
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Cited by 4 (0 self)
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We prove that every embedding of K 10 in R 3 contains a nonsplit link of three components.
COUNTING LINKS IN COMPLETE GRAPHS
, 2006
"... Abstract. We find the minimal number of links in an embedding of any complete kpartite graph on 7 vertices (including K7, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete kpartite graphs on 8 vertices. We also look ..."
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Cited by 4 (0 self)
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Abstract. We find the minimal number of links in an embedding of any complete kpartite graph on 7 vertices (including K7, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete kpartite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings. 1.