Results 1  10
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13
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 Colin de Verdière graph parameter
, 1997
"... In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless emb ..."
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Cited by 16 (1 self)
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In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.
Steinitz representations of polyhedra and the Colin de Verdière number
 J. COMB. THEORY, SER. B
, 2000
"... We show that the Steinitz representations of 3connected planar graphs are correspond, in a well described way, to Colin de Verdière matrices of such graphs. ..."
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Cited by 10 (1 self)
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We show that the Steinitz representations of 3connected planar graphs are correspond, in a well described way, to Colin de Verdière matrices of such graphs.
Geometric Representations of Graphs
 IN PAUL ERDÖS, PROC. CONF
, 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
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Cited by 8 (0 self)
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The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
The Stable Set Problem and the LiftandProject Ranks of Graphs
, 2002
"... We study the liftandproject procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the o ..."
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Cited by 5 (2 self)
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We study the liftandproject procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N 0 , N , or N+ rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N 0  or the N  rank. Hence we obtain the ranks for these graphs, and we also present some graphminor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N+ ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N+ rank of a graph.
On the Relation Between Two MinorMonotone Graph Parameters
, 1997
"... We prove that for each graph (G) (G) + 2, where and are minormonotone graph invariants introduced by Colin de Verdi'ere [3] and van der Holst, Laurent and Schrijver [5]. It is also shown that a graph G exists with (G) ! (G). The graphs G with maximal planar complement and (G) = jV (G)j ..."
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Cited by 4 (0 self)
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We prove that for each graph (G) (G) + 2, where and are minormonotone graph invariants introduced by Colin de Verdi'ere [3] and van der Holst, Laurent and Schrijver [5]. It is also shown that a graph G exists with (G) ! (G). The graphs G with maximal planar complement and (G) = jV (G)j \Gamma 4, characterised by Kotlov, Lov'asz and Vempala, are shown to be forbidden minors for fH j (H) ! jV (G)j \Gamma 4g. Introduction Given a graph G = (V; E) without loops or multiple edges, define OG as the collection of realvalued symmetric V \Theta V matrices M = (m ij ) satisfying 1. if ij 2 E, then m ij ! 0, and 2. if ij 62 E and i 6= j, then m ij = 0. There is no restriction on the diagonal entries. The elements of OG are sometimes called discrete Schrodinger operators. A matrix M 2 OG satisfies the Strong Arnol'd Hypothesis, SAH for short, if there is no nonzero symmetric matrix X = (x ij ) such that MX = 0, and such that x ij = 0 whenever i = j or ij 2 E. By i (M) we denote ...