25 research outputs found
Pitfalls of using PQ-trees in Automatic Graph Drawing
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
A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of 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 -level planar. For the final diagram the removed edges are reinserted into a -level planar drawing. Hence, i nstead of considering the -level crossing minimization problem, we suggest solv ing the -level planarization problem. In this paper we address the case . 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 . 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
The stack number of a directed acyclic graph is the minimum for which
there is a topological ordering of and a -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 -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
In this paper, we introduce and study multilevel planarity, a generalization of upward planarity and level planarity. Let be a directed graph and let be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of is a planar drawing of such that for each vertex its -coordinate is in , nd each edge is drawn as a strictly -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
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 of directions symmetric with respect to the
origin. Our results on Ordered Level Planarity imply -hardness for any
with even if the given graph is a matching. Katz, Krug, Rutter and
Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where
contains precisely the horizontal and vertical directions, can be solved in
polynomial time [GD'09]. Our results imply that this is incorrect unless
. 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
Kreuzungen in Cluster-Level-Graphen
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