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Convex drawings of Planar Graphs and the Order Dimension of 3Polytopes
 ORDER
, 2000
"... We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. The d ..."
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Cited by 33 (13 self)
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We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.
Empty Rectangles and Graph Dimension
, 2006
"... Abstract We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axisaligned rectangles. The maximum number of edges of such a graph on n points is shown to be ⌊ 1 4 n2 + n − 2⌋. This number also has other interpretations: • It is the maximum ..."
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Cited by 3 (2 self)
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Abstract We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axisaligned rectangles. The maximum number of edges of such a graph on n points is shown to be ⌊ 1 4 n2 + n − 2⌋. This number also has other interpretations: • It is the maximum number of edges of a graph of dimension [3 ↕↕4], i.e., of a graph with a realizer of the form π1, π2, π1, π2. • It is the number of 1faces in a special Scarf complex. The last of these interpretations allows to deduce the maximum number of empty axisaligned rectangles spanned by 4element subsets of a set of n points. Moreover, it follows that the extremal point sets for the two problems coincide. We investigate the maximum number of of edges of a graph of dimension [3 ↕ 4], i.e., of a graph with a realizer of the form π1, π2, π3, π3. This maximum is shown to be 1 4 n2 + O(n). Box graphs are defined as the 3dimensional analog of rectangle graphs. The maximum number of edges of such a graph on n points is shown to be 7 16 n2 + o(n 2). Mathematics Subject Classifications (2000). 05C10, 68R10, 06A07. 1
Orthogonal Surfaces and their CPorders
, 2007
"... Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with c ..."
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Cited by 2 (2 self)
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Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and 3polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which nongeneric orthogonal surfaces have a polytopal structure. We review the state of knowledge of the 3dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cporders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cporders can lack key properties of face lattices. We investigate extra requirements which may help to have cporders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.
Highdimensional orthogonal surfaces
, 2006
"... Abstract. Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich struct ..."
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Cited by 1 (1 self)
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Abstract. Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and 3polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which nongeneric orthogonal surfaces have a polytopal structure. We study characteristic points and the cporders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cporders can lack key properties of face lattices. We investigate extra requirements which may help to have cporders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces. Mathematics Subject Classifications (2000). 05C62, 06A07, 52B05, 68R10. 1
DIMENSION AND STRUCTURE FOR A POSET OF GRAPH MINORS
"... Abstract. Given a graph G with labeled vertices, define MP(G), the labeled minorposet of G, to be the poset whose elements are the minors of G with G1 ≤ G2 if and only if G1 ≼ G2. In this paper we study the dimension this poset, which is a minormonotone graph parameter. We provide important struct ..."
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Abstract. Given a graph G with labeled vertices, define MP(G), the labeled minorposet of G, to be the poset whose elements are the minors of G with G1 ≤ G2 if and only if G1 ≼ G2. In this paper we study the dimension this poset, which is a minormonotone graph parameter. We provide important structural results which yield nontrivial bounds on this parameter for cycles, complete graphs, trees, and the minorclosed class of graphs which exclude K2,4minors. We also state a conjecture that characterizes of the class of K2,tminor free graphs. Lastly, we consider two multigraph models and provide direction for future research. 1.
The Order Dimension of Planar Maps Revisited
"... Abstract. 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 fac ..."
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Abstract. 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 planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyderpaths and Schnyderregions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(PM)) ≤ 4. This may be the first result in the area that is obtained without using the tools introduced by Schnyder.