25 research outputs found

    Pitfalls of using PQ-trees in Automatic Graph Drawing

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    A number of erroneous attempts involving PQ-trees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particular, we examine erroneous usage of the PQ-tree data structure in algorithms for computing maximal planar subgraphs and an algorithm for testing leveled planarity of leveled directed acyclic graphs with several sources and sinks

    An alternative method to crossing minimization on hierarchical graphs

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    A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of kk levels and then, as a second step, to permute the verti ces within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, removing the minimal number of edges such that the resulting graph is kk-level planar. For the final diagram the removed edges are reinserted into a kk-level planar drawing. Hence, i nstead of considering the kk-level crossing minimization problem, we suggest solv ing the kk-level planarization problem. In this paper we address the case k=2k=2. First, we give a motivation for our appro ach. Then, we address the problem of extracting a 2-level planar subgraph of maximum we ight in a given 2-level graph. This problem is NP-hard. Based on a characterizatio n of 2-level planar graphs, we give an integer linear programming formulation for the 2-level planarization problem. Moreover, we define and investigate the polytop e \2LPS(G) associated with the set of all 2-level planar subgraphs of a given 2 -level graph GG. We will see that this polytope has full dimension and that the i nequalities occuring in the integer linear description are facet-defining for \2L PS(G). The inequalities in the integer linear programming formulation can be separated in polynomial time, hence they can be used efficiently in a branch-and-cut method fo r solving practical instances of the 2-level planarization problem. Furthermore, we derive new inequalities that substantially improve the quality of the obtained solution. We report on extensive computational results

    Directed Acyclic Outerplanar Graphs Have Constant Stack Number

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    The stack number of a directed acyclic graph GG is the minimum kk for which there is a topological ordering of GG and a kk-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [GD 2021] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed HH-partitions, which might be of independent interest. We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by N\"ollenburg and Pupyrev [arXiv:2107.13658, 2021]

    Multilevel Planarity

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    In this paper, we introduce and study multilevel planarity, a generalization of upward planarity and level planarity. Let G=(V,E)G = (V, E) be a directed graph and let ℓ:V→P(Z)\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 GG is a planar drawing of GG such that for each vertex v∈Vv\in V its yy-coordinate y(v)y(v) is in ℓ(v)\ell(v), nd each edge is drawn as a strictly yy-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 NP-complete even in very restricted cases

    Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

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    We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a polygonal path composed of line segments with two adjacent directions from a given set SS of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply NPNP-hardness for any SS with ∣S∣≥4|S|\ge 4 even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where SS contains precisely the horizontal and vertical directions, can be solved in polynomial time [GD'09]. Our results imply that this is incorrect unless P=NPP=NP. Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer and \v{S}tefankovi\v{c}. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati and Roselli.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing

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    Kreuzungen in Cluster-Level-Graphen

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    Clustered graphs are an enhanced graph model with a recursive clustering of the vertices according to a given nesting relation. This prime technique for expressing coherence of certain parts of the graph is used in many applications, such as biochemical pathways and UML class diagrams. For directed clustered graphs usually level drawings are used, leading to clustered level graphs. In this thesis we analyze the interrelation of clusters and levels and their influence on edge crossings and cluster/edge crossings.Cluster-Graphen sind ein erweitertes Graph-Modell mit einem rekursiven Clustering der Knoten entsprechend einer gegebenen Inklusionsrelation. Diese bedeutende Technik um Zusammengehörigkeit bestimmter Teile des Graphen auszudrücken wird in vielen Anwendungen benutzt, etwa biochemischen Reaktionsnetzen oder UML Klassendiagrammen. Für gerichtete Cluster-Graphen werden üblicherweise Level-Zeichnungen verwendet, was zu Cluster-Level-Graphen führt. Diese Arbeit analysiert den Zusammenhang zwischen Clustern und Level und deren Auswirkungen auf Kantenkreuzungen und Cluster/Kanten-Kreuzungen
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