Upward Planar Research Articles

Upward planarity testing of biconnected outerplanar DAGs solves partition

We show an O(n)-time reduction from the problem of testing whether a multiset of positive integers can be partitioned into two multisets so that the sum of the integers in each multiset is equal to n/2 to the problem of testing whether an n-vertex biconnected outerplanar DAG admits an upward planar drawing. This constitutes the first barrier to the existence of efficient algorithms for testing the upward planarity of DAGs with no large triconnected minor.We also show a result in the opposite direction. Suppose that partitioning a multiset of positive integers into two multisets so that the sum of the integers in each multiset is n/2 can be solved in f(n) time. Let G be an n-vertex biconnected outerplanar DAG and e be an edge incident to the outer face of an outerplanar drawing of G. Then it can be tested in O(f(n)) time whether G admits an upward planar drawing with e on the outer face.

Bitonic st-Orderings for Upward Planar Graphs: Splits and Bends in the Variable Embedding Scenario

Bitonic st-orderings for st-planar graphs were introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge in polynomial area. For an st-planar graph that does not admit a bitonic st-ordering, one may split certain edges such that for the resulting graph such an ordering exists. Since each split is interpreted as a bend, one is usually interested in splitting as few edges as possible. While this optimization problem admits a linear-time algorithm in the fixed embedding setting, it remains open in the variable embedding setting. We close this gap in the literature by providing a linear-time algorithm that optimizes over all embeddings of the input st-planar graph. The best-known lower bound on the number of required splits of an st-planar graph with n vertices is n-3. However, it is possible to compute a bitonic st-ordering without any split for the st-planar graph obtained by reversing the orientation of all edges. In terms of upward planar polyline drawings in polynomial area, the former translates into n-3 bends, while the latter into no bends. We show that this idea cannot always be exploited by describing an st-planar graph that needs at least n-5 splits in both orientations. We provide analogous bounds for graphs with small degree. Finally, we further investigate the relationship between splits in bitonic st-orderings and bends in upward planar polyline drawings with polynomial area, by providing bounds on the number of bends in such drawings.

Planar Rooted Phylogenetic Networks.

A rooted phylogenetic network is a directed acyclic graph with a single root, whose sinks correspond to a set of species. As such networks are useful for representing the evolution of species that have undergone reticulate evolution, there has been great interest in developing the theory behind and algorithms for constructing them. However, unlike evolutionary trees, these networks can be highly non-planar, which can make them difficult to visualise and interpret. Here we investigate properties of planar rooted phylogenetic networks and algorithms for deciding whether or not rooted networks have certain special planarity properties. In particular, we introduce three natural subclasses of planar rooted phylogenetic networks and show that they form a hierarchy. In addition, for the well-known level- k networks, we show that level-1, -2, -3 networks are always outer, terminal, and upward planar, respectively, and that level-4 networks are not necessarily planar. Finally, we show that a regular network is terminal planar if and only if it is pyramidal. Our results make use of the highly developed field of planar digraphs, and we believe that the link between phylogenetic networks and planar graphs should prove useful in future for developing new approaches to both construct and visualise phylogenetic networks.

Upward Planar Drawings with Three and More Slopes

The slope number of a graph $G$ is the smallest number of slopes needed for the segments representing the edges in any straight-line drawing of $G$. It serves as a measure of the visual complexity of a graph drawing. Several bounds on the slope number for particular graph classes have been established, both in the planar and the non-planar setting. Moreover, the slope number can also be defined for directed graphs and upward planar drawings. We study upward planar straight-line drawings that use only a constant number of slopes. In particular, for a fixed number $k$ of slopes, we are interested in whether a given directed graph $G$ with maximum in- and outdegree at most $k$ admits an upward planar $k$-slope drawing. We investigate this question both in the fixed and the variable embedding scenario. We show that this problem is in general NP-hard to decide for outerplanar graphs ($k = 3$) and planar graphs ($k \ge 3$). On the positive side, we can decide whether a given cactus graph admits an upward planar $k$-slope drawing and, in the affirmative, construct such a drawing in FPT time with parameter $k$. Furthermore, we can determine the minimum number of slopes required for a given tree in linear time and compute the corresponding drawing efficiently.

Upward planar drawings with two slopes

In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a drawing and, if so, how to construct it. For the fixed embedding scenario, we give a simple characterisation and a linear-time construction by adopting algorithms from orthogonal drawings. For the variable embedding scenario, we describe a linear-time algorithm for single-source digraphs, a quartic-time algorithm for series-parallel digraphs, and a fixed-parameter tractable algorithm for general digraphs. For the latter two classes, we make use of SPQR-trees and the notion of upward spirality. As an application of this drawing style, we show how to draw an upward planar phylogenetic network with two slopes such that all leaves lie on a horizontal line.

Radial Level Planarity with Fixed Embedding

We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems.

Multilevel Planarity

In this paper, we introduce and study multilevel planarity, a generalization of upward planarity and level planarity. Let $G = (V, E)$ be a directed graph and let $\ell: V \to \mathcal P(\mathbb Z)$ be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of $G$ is a planar drawing of $G$ such that for each vertex $v\in V$ its $y$-coordinate $y(v)$ is in $\ell(v)$, and each edge is drawn as a strictly $y$-monotone curve. We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source and of oriented cycles. Complementing these algorithmic results, we show that multilevel-planarity testing is $\textsf{NP}$-complete even in very restricted cases.

Extending upward planar graph drawings

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing.We show the following results.–First, we prove that the Upward Planarity Extension problem is NP-complete, even if G has a prescribed upward embedding, the vertex set of H coincides with the one of G, and H contains no edge.–Second, we show that the Upward Planarity Extension problem can be solved in O(nlog⁡n) time if G is an n-vertex upward planar st-graph. This result improves upon a known O(n2)-time algorithm, which however applies to all n-vertex single-source upward planar graphs.–Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which the underlying graph of G is a path or a cycle, G has a prescribed upward embedding, H contains no edges, and no two vertices share the same y-coordinate in ΓH.

Upward Planar Morphs

We prove that, given two topologically-equivalent upward planar straight-line drawings of an n-vertex directed graph G, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of O(1) morphing steps if G is a reduced planar st-graph, O(n) morphing steps if G is a planar st-graph, O(n) morphing steps if G is a reduced upward planar graph, and $$O(n^2)$$ morphing steps if G is a general upward planar graph. Further, we show that $$\varOmega (n)$$ morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an n-vertex path.

Combinatorial Characterization of Upward Planarity

We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.

Dynamic Upward Planarity Testing of Single Source Embedded Digraphs

Dynamic Upward Planarity Testing of Single Source Embedded Digraphs

Strip Planarity Testing for Embedded Planar Graphs

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph G(V, E) and a function $$\gamma :V \rightarrow \{1,2,\dots ,k\}$$ and asks whether a planar drawing of G exists such that each edge is represented by a curve that is monotone in the y-direction and, for any $$u,v\in V$$ with $$\gamma (u)<\gamma (v)$$ , it holds that $$y(u)<y(v)$$ . The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. Most notably, we provide a polynomial-time reduction from strip planarity testing to clustered planarity. We show that the strip planarity testing problem is polynomial-time solvable if G has a prescribed combinatorial embedding.

Upward Planarity Testing in Practice

A directed acyclic graph (DAG) is upward planar if it can be drawn without any crossings while all edges—when following them in their direction—are drawn with strictly monotonously increasing y -coordinates. Testing whether a graph allows such a drawing is known to be NP-complete, and while the problem is polynomial-time solvable for special graph classes, there is not much known about solving the problem for general graphs in practice . The only attempt so far has been a branch-and-bound algorithm over the graph’s triconnectivity structure, which was able to solve small graphs. Furthermore, there are some known FPT algorithms to deal with the problem. In this article, we propose two fundamentally different approaches based on the seemingly novel concept of ordered embeddings and on the concept of a Hanani-Tutte-type characterization of monotone drawings. In both approaches, we model the problem as special SAT instances, that is, logic formulae for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. For the first time, we give an extensive experimental comparison between virtually all known approaches to the problem. To this end, we also investigate implementation issues and different variants of the known algorithms as well as of our SAT approaches and evaluate all algorithms on real-world as well as on constructed instances. We also give a detailed performance study of the novel SAT approaches. We show that the SAT formulations outperform all known approaches for graphs with up to 400 edges. For even larger graphs, a modified branch-and-bound algorithm becomes competitive.

Upward planar graphs and their duals

We consider upward planar drawings of directed graphs in the plane (UP), and on standing (SUP) and rolling cylinders (RUP). In the plane and on the standing cylinder the edge curves are monotonically increasing in y-direction. On the rolling cylinder they wind unidirectionally around the cylinder. There is a strict hierarchy of classes of upward planar graphs: UP⊂SUP⊂RUP.In this paper, we show that rolling and standing cylinders switch roles when considering an upward planar graph and its dual. In particular, we prove that a strongly connected graph is RUP if and only if its dual is a SUP dipole. A dipole is an acyclic graph with a single source and a single sink. All RUP graphs are characterized in terms of their duals using generalized dipoles. Moreover, we obtain a characterization of the primals and duals of wSUP graphs which are upward planar graphs on the standing cylinder and allow for horizontal edge curves.

Monotone Simultaneous Embeddings of Upward Planar Digraphs

We study monotone simultaneous embeddings of upward planar digraphs, which are simultaneous embeddings where the drawing of each digraph is upward planar, and the directions of the upwardness of different graphs can differ. We first consider the special case where each digraph is a directed path. In contrast to the known result that any two directed paths admit a monotone simultaneous embedding, there exist examples of three paths that do not admit such an embedding for any possible choice of directions of monotonicity. We prove that if a monotone simultaneous embedding of three paths exists then it also exists for any possible choice of directions of monotonicity. We provide a polynomial-time algorithm that, given three paths, decides whether a monotone simultaneous embedding exists and, in the case of existence, also constructs such an embedding. On the other hand, we show that already for three paths, any monotone simultaneous embedding might need a grid whose size is exponential in the number of vertices. For more than three paths, we present a polynomial-time algorithm that, given any number of paths and predefined directions of monotonicity, decides whether the paths admit a monotone simultaneous embedding with respect to the given directions, including the construction of a solution if it exists. Further, we show several implications of our results on monotone simultaneous embeddings of general upward planar digraphs. Finally, we discuss complexity issues related to our problems. ∗Research supported by the ESF EUROCORES programme EuroGIGA ComPoSe, Austrian Science Fund (FWF): I 648-N18 and grant EUI-EURC-2011-4306. T.H. supported by the Austrian Science Fund (FWF): P23629-N18 ‘Combinatorial Problems on Geometric Graphs’. Work partly done while A.P. was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. †Institute for Software Technology, Graz University of Technology, Austria, [oaich|thackl|apilz|bvogt]@ist.tugraz.at. ‡Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany, sarah.lutteropp@student.kit.edu, mched@iti.uka.de. A part of this research has been accomplished when the authors were visiting Graz University of Technology.

Computing upward topological book embeddings of upward planar digraphs

We describe a unified approach for studying book, point-set, and simultaneous embeddability problems of upward planar digraphs. The approach is based on a linear time strategy to compute an upward planar drawing of an upward planar digraph such that all vertices are collinear. Besides having impact in relevant application domains of graph drawing and computational geometry, the presented results open new research directions in the area of upward planarity with constraints of the positions of the vertices.

On the number of upward planar orientations of maximal planar graphs

We consider the problem of determining the maximum and the minimum number of upward planar orientations a maximal planar graph can have. We show that every n-vertex maximal planar graph has at least Ω⁎(1.189n) and at most O⁎(4n) upward planar orientations. Moreover, we show that there exist n-vertex maximal planar graphs having O⁎(2n) upward planar orientations and n-vertex maximal planar graphs having Ω⁎(2.599n) upward planar orientations. Further, we present bounds for the maximum and the minimum number of acyclic orientations a maximal planar graph can have.

Upward and quasi-upward planarity testing of embedded mixed graphs

Mixed graphs have both directed and undirected edges and have received considerable attention in the literature. We study two upward planarity testing problems for embedded mixed graphs, give some NP-hardness results, and describe Integer Linear Programming techniques to solve them. Experiments show the efficiency of our approach.

On the Upward Planarity of Mixed Plane Graphs

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 $\vec G$ . 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.

Reprint of: Upward planar embedding of an n-vertex oriented path on [formula omitted] points

We prove that every n-vertex oriented path admits an upward planar embedding on every general set of (n−1)2+1 points on the plane. This result improves the previously known upper bound which is exponential in the number of switches of the given oriented path (Angelini et al. 2010) [1].