Combinatorial Characterization of Upward Planarity (original) (raw)
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Journal of Graph Theory, 2005
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 Hoşten 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.
arXiv: Combinatorics, 2019
A planar order is a special linear extension of the edge poset (partially ordered set) of a processive plane graph. The definition of a planar order makes sense for any finite poset and is equivalent to the one of a conjugate order. Here it was proved that there is a planar order on the vertex poset of a processive planar graph naturally induced from the planar order of its edge poset.
Planar graphs and poset dimension
Order, 1989
We view the incidence relation of a graph G=(I: E) as an order relation on ~ts vertices and edges, i.e. a <6 b if and only ifa is a vertex and b is an edge incident on a. This leads to the definition of the order-dimension of G as the minimum number of total orders on I'~ E whose intersection is <G. Our main result ~s the characterization of planar graphs as the graphs whose order-dimension does not exceed three. Strong versions of several known properties of planar graphs are ~mphed by this characterizahon These properhes include: each planar graph has arboricn) at most three and each planar graph has a plane embedding ~hose edges are straight line segments. A nice feature of this embedding ~s that the coordinates of the vertices have a purely combmatoriaI meaning.
Adjacency posets of planar graphs
Discrete Mathematics, 2010
In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertex-face poset of such a graph.
Bipartite Graphs, Upward Drawings, and Planarity
Information Processing Letters, 1990
We show that a bipartite digraph admits an upward drawing. i.e.. a planar drawing with the additional constraint that all the edges flow in the same direction if and only if it is planar. This result finds applications both in the field of automatic graph layout and in the field of ordered sets.
On the Upward Planarity of Mixed Plane Graphs
Journal of Graph Algorithms and Applications, 2014
A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges and a set of undirected edges. An orientation of a mixed plane graph G is an assignment of directions to the undirected edges of G resulting in a directed plane graph G. In this paper, we study the computational complexity of testing whether a given mixed plane graph G is upward planar, i.e., whether it admits an orientation resulting in a directed plane graph G such that G admits a planar drawing in which each edge is represented by a curve monotonically increasing in the y-direction according to its orientation. Our contribution is threefold. First, we show that the upward planarity testing problem is solvable in cubic time for mixed outerplane graphs. Second, we show that the problem of testing the upward planarity of mixed plane graphs reduces in quadratic time to the problem of testing the upward planarity of mixed plane triangulations. Third, we exhibit linear-time testing algorithms for two classes of mixed plane triangulations, namely mixed plane 3-trees and mixed plane triangulations in which the undirected edges induce a forest.
The dimension of planar posets
Journal of Combinatorial Theory, Series B, 1977
A partially ordered set (poset) is planar if it has a planar Hasse diagram. The dimension of a bounded planar poset is at most two. We show that the dimension of a planar poset having a greatest lower bound is at most three. We also construct four-dimensional planar posets, but no planar poset with dimension larger than four is known. A poset is called a tree if its Hasse diagram is a tree in the graph-theoretic sense. We show that the dimension of a tree is at most three and give a forbidden subposet characterization of two-dimensional trees.
The Dimension of Posets with Planar Cover Graphs
Graphs and Combinatorics, 2014
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.
Upward Embeddings and Orientations of Undirected Planar Graphs
Journal of Graph Algorithms and Applications, 2003
An upward embedding of an embedded planar graph specifies, for each vertex v, which edges are incident on v "above" or "below" and, in turn, induces an upward orientation of the edges from bottom to top. In this paper we characterize the set of all upward embeddings and orientations of an embedded planar graph by using a simple flow model, which is related to that described by Bousset [3] to characterize bipolar orientations. We take advantage of such a flow model to compute upward orientations with the minimum number of sources and sinks of 1-connected embedded planar graphs. We finally devise a new algorithm for computing visibility representations of 1-connected planar graphs using our theoretic results.