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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 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.
THE DIMENSION OF POSETS WITH PLANAR COVER GRAPHS
"... Abstract. Kelly showed that there exist planar posets of arbitrarily large dimension, and Streib and Trotter showed that the dimension of a poset with a planar cover graph is bounded in terms of its height. Here we continue the study of conditions that bound the dimension of posets with planar cover ..."
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Abstract. Kelly showed that there exist planar posets of arbitrarily large dimension, and Streib and Trotter showed that the dimension of a poset with a planar cover graph is bounded in terms of its height. Here we continue the study of conditions that bound the dimension of posets with planar cover graphs. We show that if P is poset with a planar comparability graph, then the dimension of P is at most four. We also show that if P has an outerplanar cover graph, then the dimension of P is at most four. Finally, if P has an outerplanar cover graph and the height of P is two, then the dimension of P is at most three. These three inequalities are all best possible. 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.