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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 10 (7 self)
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↕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
The order dimension of planar maps
 SIAM J. DISCRETE MATH
, 1997
"... This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3connected planar map M, ordered by inclusion, ..."
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Cited by 13 (5 self)
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This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3connected planar map M, ordered by inclusion
The Order Dimension of Planar Maps Revisited
"... Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the BrightwellTrotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a ..."
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Cited by 1 (1 self)
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Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the BrightwellTrotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a
The polytope of noncrossing graphs on a planar point set
, 2003
"... For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “noncrossing marked graphs” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of noncr ..."
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Cited by 11 (4 self)
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crossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni + n − 3 where ni is the number of points of A in the interior of conv(A). The vertices of this polytope are all the pseudotriangulations of A, and the edges
On the Order Dimension of Outerplanar Maps
, 2007
"... Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertexedgeface poset PM of a planar map M is at most four. In this paper we investigate cases where dim(PM) ≤ 3 and also where dim(QM) ≤ 3; here QM denotes the vertexface pos ..."
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Cited by 3 (3 self)
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Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertexedgeface poset PM of a planar map M is at most four. In this paper we investigate cases where dim(PM) ≤ 3 and also where dim(QM) ≤ 3; here QM denotes the vertex
Geometric Containment Orders: A Survey
 ORDER
, 1999
"... A partially ordered set (X, ≺) is a geometric containment order of a particular type if there is a mapping from X into similarly shaped objects in a finitedimensional Euclidean space that preserves ≺ by proper inclusion. This survey describes most of what is presently known about geometric containm ..."
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Cited by 1 (0 self)
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containment orders. Highlighted shapes include angular regions, convex polygons and circles in the plane, and spheres of all dimensions. Containment orders are also related to incidence orders for vertices, edges and faces of graphs, hypergraphs, planar graphs and convex polytopes. Three measures of poset
Akademisk avhandling för teknisk doktorsexamen vid
, 1994
"... mcmxciv This thesis deals with combinatorics in connection with Coxeter groups, finitely generated but not necessarily finite. The representation theory of groups as nonsingular matrices over a field is of immense theoretical importance, but also basic for computational group theory, where the group ..."
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of the thesis can be summarized as follows. • We prove that for all Coxeter graphs constructed from an npath of unlabelled edges by adding a new labelled edge and a new vertex (sometimes two new edges and vertices), there is a permutational representation of the corresponding group. Group elements correspond