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Building blocks of upward planar digraphs
 Proc. GD’04, volume 3383 of LNCS
, 2005
"... The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if ..."
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The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if its blocks admit upward planar drawings with certain properties. We also show how to test if a block of a digraph admits an upward planar drawing with the aforementioned properties.
Upward Planarity Testing via SAT
"... Abstract. A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing ycoordinates. The problem to decide whether a graph is upward planar or not is NPcomplete in general, and while special graph classes a ..."
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Abstract. A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing ycoordinates. The problem to decide whether a graph is upward planar or not is NPcomplete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branchandbound algorithm over the graph’s triconnectivity structure which was able to solve sparse graphs. In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively. 1
On the Complexity of Finding . . .
, 2007
"... Upward planarity is a widely investigated topic in graph theory and graph drawing. A planar digraph is said to be upward planar if it admits a planar drawing where all edges are drawn as curves monotonically increasing in the upward direction. It is known that testing whether a planar digraph is upw ..."
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Upward planarity is a widely investigated topic in graph theory and graph drawing. A planar digraph is said to be upward planar if it admits a planar drawing where all edges are drawn as curves monotonically increasing in the upward direction. It is known that testing whether a planar digraph is upward planar is NPComplete in general. However, the upward planarity testing problem can be solved in polynomial time if the planar embedding of the digraph is fixed. This motivates the following question: Given an embedded planar digraph G, how is difficult to find an embedding preserving subgraph of G that is upward planar and whose number of edges is maximum? We prove that this problem is NPHard. This negative result motivates the study of polynomialtime algorithms for the computation of maximal (not necessarily maximum) upward planar subgraphs as well as the design of exact algorithms that perform well in practice. As a consequence of our proof technique, we also show that the problem of extracting a maximum bimodal subgraph from an embedded planar digraph is NPHard.
Upward Planar Drawings . . .
, 2006
"... In this paper we present a new characterization of switchregular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popula ..."
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In this paper we present a new characterization of switchregular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popular approach described by Bertolazzi, Di Battista, Liotta, and Mannino, our algorithm computes drawings that are significantly better in terms of total edge length and aspect ratio, especially for lowdensity digraphs. Also, we experimentally prove that the running time of the drawing process is reduced in most cases.
Communicated by:
, 2010
"... Upward planar drawings of digraphs are crossing free drawings where all edges flow in the upward direction. The problem of deciding whether a digraph admits an upward planar drawing is called the upward planarity testing problem, and it has been widely studied in the literature. In this paper we inv ..."
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Upward planar drawings of digraphs are crossing free drawings where all edges flow in the upward direction. The problem of deciding whether a digraph admits an upward planar drawing is called the upward planarity testing problem, and it has been widely studied in the literature. In this paper we investigate a new upward planarity testing problem, that is, deciding whether a digraph admits an upward planar drawing having some special topological properties: such a drawing is called switchregular. Switchregularupwardplanardrawings havepractical algorithmic impacts in several graph drawing applications. We provide characterizations for the class of directed trees that admit a switchregular upward planar drawing. Based on these characterizations we describe an optimal lineartime testing and embedding algorithm. Submitted: