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34
Posets and planar graphs
 JOURNAL OF GRAPH THEORY
, 2000
"... Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [t − 1 ↕t] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [2↕3]. ..."
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Cited by 7 (6 self)
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Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [t − 1 ↕t] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [2↕3]. This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3. • The largest n for which the dimension of the complete graph Kn is at most [t − 1↕t] is the number of antichains in the lattice of all subsets of a set of size t − 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind’s problem. This result extends work of Ho¸sten and Morris [14]. The main results are enriched by background material which links to a line of reserch in extremal graph theory which was stimulated by a problem posed by G. Agnarsson: Find the maximum number of edges in a graph on n nodes with dimension at most t.
Algebraic shifting and fvector theory
, 2007
"... This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in f ..."
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This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in fvector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in fvector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.
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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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.
On the null space of a Colin de Verdière matrix
"... Let G = (V; E) be a 3connected planar graph, with V = f1; : : : ; ng. Let M = (m i;j ) be a symmetric n \Theta n matrix with exactly one negative eigenvalue (of multiplicity 1), such that for i; j with i 6= j, if i and j are adjacent then m i;j ! 0 and if i and j are nonadjacent then m i;j = 0, and ..."
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Let G = (V; E) be a 3connected planar graph, with V = f1; : : : ; ng. Let M = (m i;j ) be a symmetric n \Theta n matrix with exactly one negative eigenvalue (of multiplicity 1), such that for i; j with i 6= j, if i and j are adjacent then m i;j ! 0 and if i and j are nonadjacent then m i;j = 0, and such that M has rank n \Gamma 3. Then the null space ker M of M gives an embedding of G in S 2 as follows: Let a; b; c be a basis of ker M , and for i 2 V let OE(i) := (a i ; b i ; c i ) T ; then OE(i) 6= 0, and /(i) := OE(i)=kOE(i)k embeds V in S 2 such that connecting, for any two adjacent vertices i; j, the points /(i) and /(j) by a shortest geodesic on S 2 , gives a proper embedding of G in S 2 . This applies to the matrices associated with the parameter (G) introduced by Y. Colin de Verdi`ere.
On Embeddability and Stresses of Graphs
, 2007
"... Gluck [6] has proven that triangulated 2spheres are generically 3rigid. Equivalently, planar graphs are generically 3stress free. We show that already the K5minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2minor free graph is generically rstress free for 1 ..."
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Gluck [6] has proven that triangulated 2spheres are generically 3rigid. Equivalently, planar graphs are generically 3stress free. We show that already the K5minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2minor free graph is generically rstress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) Some further extensions are discussed. 1 Introduction and
Discrete and Continuous: Two sides of the same?
"... How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our science are quite different. But on a deeper level, there is a more solid connection than meets the eye. ..."
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How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our science are quite different. But on a deeper level, there is a more solid connection than meets the eye.
Open Problems
, 1991
"... . Graph minors is a field that has motivated numerous investigations in discrete mathematics and computers science, a fact demonstrated by the variety of papers and open problems appearing in this volume. This section summarizes those problems which were submitted by participants of the conference f ..."
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. Graph minors is a field that has motivated numerous investigations in discrete mathematics and computers science, a fact demonstrated by the variety of papers and open problems appearing in this volume. This section summarizes those problems which were submitted by participants of the conference for inclusion in this special section of the proceedings. 1. Introduction There were many open problems discussed in problem sessions, meals, excursions and other gatherings at the conference. Several of these appear in the other papers of this volume, and a few of the participants submitted problems to be included in this section. This paper is partitioned into several sections addressing a variety of problem areas: pathwidth and treewidth (Section 2), paths, cycles and independent sets (Section 3), coverings and integer flows (Section 4), wellquasiordering (Section 5), geometry and topology (Section 6), logic (Section 7), and disjoint paths (Section 8). Each subsection focuses on prese...
Graph planarity and related topics
 Graph Drawing (Proc. GD ’99), volume 1731 of LNCS
, 1999
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