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28
The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey
, 2007
"... The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ̸ = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minim ..."
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Cited by 41 (17 self)
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The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ̸ = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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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
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.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound t ..."
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Cited by 8 (0 self)
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In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
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.
Recent Excluded Minor Theorems
 SURVEYS IN COMBINATORICS, LMS LECTURE NOTE SERIES
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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Cited by 3 (1 self)
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
The van Kampen obstruction and its relatives
, 2009
"... We review a cochainfree treatment of the classical van Kampen obstruction ϑ to embeddability of an npolyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.P. Dax. The following results ar ..."
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Cited by 2 (0 self)
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We review a cochainfree treatment of the classical van Kampen obstruction ϑ to embeddability of an npolyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.P. Dax. The following results are obtained. • The mod2 reduction of ϑ is incomplete, which answers a question of Sarkaria. • An odddimensional analogue of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of the given npolyhedron in R2n+1. • A “blown up ” 1parameter version of ϑ is a universal type 1 invariant of singular knots, i.e. knots in R3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integervalued type 1 invariant of singular knots admits an integral arrow diagram ( = Polyak–Viro) formula. • Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, nonembeddable in a given Rm, m ≥ 3(n+1), contains a subset X 2 such that no map N → Rm sends X and N \ X to disjoint sets. • We elaborate on McCrory’s analysis of the Zeeman spectral sequence to geometrically characterize “kcoconnected and locally kcoconnected ” polyhedra, which we embed in R2n−k for k < n−3 extending the Penrose–Whitehead–Zeeman theorem.