Results 1  10
of
25
An Alternative Method to Crossing Minimization on Hierarchical Graphs
 SIAM J. Optimization
, 1997
"... . A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, remo ..."
Abstract

Cited by 28 (0 self)
 Add to MetaCart
. A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices 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 klevel planar. For the final diagram the removed edges are reinserted into a klevel planar drawing. Hence, instead of considering the klevel crossing minimization problem, we suggest solving the klevel planarization problem. In this paper we address the case k = 2. First, we give a motivation for our approach. Then, we address the problem of extracting a 2level planar subgraph of maximum weight in a given 2level graph. This problem is NPhard. Based on a characterization of 2level planar graphs, we give an integer linear programming formulation for the 2level planarization problem. Moreover, we define and investigate t...
Stack And Queue Layouts Of Directed Acyclic Graphs: Part I
, 1996
"... . Stack layouts and queue layouts of undirected graphs have been used to model problems in fault tolerant computing and in parallel process scheduling. However, problems in parallel process scheduling are more accurately modeled by stack and queue layouts of directed acyclic graphs (dags). A stack ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
. Stack layouts and queue layouts of undirected graphs have been used to model problems in fault tolerant computing and in parallel process scheduling. However, problems in parallel process scheduling are more accurately modeled by stack and queue layouts of directed acyclic graphs (dags). A stack layout of a dag is similar to a stack layout of an undirected graph, with the additional requirement that the nodes of the dag be in some topological order. A queue layout is defined in an analogous manner. The stacknumber (queuenumber) of a dag is the smallest number of stacks (queues) required for its stack layout (queue layout). In this paper, bounds are established on the stacknumber and queuenumber of two classes of dags: tree dags and unicyclic dags. In particular, any tree dag can be laid out in 1 stack and in at most 2 queues; and any unicyclic dag can be laid out in at most 2 stacks and in at most 2 queues. Forbidden subgraph characterizations of 1queue tree dags and 1queue cycle d...
Level Planarity Testing in Linear Time
, 1999
"... A level graph G = (V; E; OE) is a directed acyclic graph with a mapping OE : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [ \Delta \Delta \Delta [V k , V j = OE \Gamma1 (j), V i " V j = ; for i 6= j, such that OE(v) OE(u) + 1 for each edge (u; v) 2 E. The level planar ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
A level graph G = (V; E; OE) is a directed acyclic graph with a mapping OE : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [ \Delta \Delta \Delta [V k , V j = OE \Gamma1 (j), V i " V j = ; for i 6= j, such that OE(v) OE(u) + 1 for each edge (u; v) 2 E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i , all v 2 V i are drawn on the line l i = f(x; k \Gamma i) j x 2 Rg, the edges are drawn monotone with respect to the vertical direction, and no edges intersect except at their end vertices. If G has a single source, the test can be performed in O(jV j) time by an algorithm of Di Battista and Nardelli [1988] that uses the PQtree data structure introduced by Booth and Lueker [1976]. PQtrees have also been proposed by Heath and Pemmaraju [1995, 1996] to test level planarity of level directed acyclic graphs with several sources and sinks. It has been shown in Jünger, Leipert, and...
Radial Level Planarity Testing and Embedding in Linear Time
 Journal of Graph Algorithms and Applications
, 2005
"... A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines an ..."
Abstract

Cited by 19 (9 self)
 Add to MetaCart
A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are level nonplanar biconnected components. Our main results are linear time algorithms for radial level planarity testing and for computing an embedding. We introduce PQRtrees as a new data structure where Rnodes and associated templates for their manipulation are introduced to deal with rings. Our algorithms extend level planarity testing and embedding algorithms which use PQtrees.
An Algorithm For Drawing A Hierarchical Graph
, 1995
"... this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1. ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1.
Characterization of unlabeled level planar trees
 14TH SYMPOSIUM ON GRAPH DRAWING (GD), VOLUME 4372 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... Consider a graph G drawn in the plane so that each vertex lies on a distinct horizontal line ℓj = {(x, j)  x ∈ R}. The bijection φ that maps the set of n vertices V to a set of distinct horizontal lines ℓj forms a labeling of the vertices. Such a graph G with the labeling φ is called an nlevel gr ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
Consider a graph G drawn in the plane so that each vertex lies on a distinct horizontal line ℓj = {(x, j)  x ∈ R}. The bijection φ that maps the set of n vertices V to a set of distinct horizontal lines ℓj forms a labeling of the vertices. Such a graph G with the labeling φ is called an nlevel graph and is said to be nlevel planar if it can be drawn with straightline edges and no crossings while keeping each vertex on its own level. In this paper, we consider the class of trees that are nlevel planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are threefold. First, we provide a complete characterization of ULP trees in terms of a pair of forbidden subtrees. Second, we show how to draw ULP trees in linear time. Third, we provide a linear time recognition algorithm for ULP trees.
Recognizing LeveledPlanar Dags in Linear Time
 In Proc. Graph Drawing, GD 1995, volume 1027 of LNCS
, 1996
"... this paper we present a linear time algorithm for the problem of determining if a given dag has a directed leveledplanar embedding. Our algorithm uses a variation of the PQtree data structure introduced by Booth and Lueker [2]. One motivation for our algorithm is that it can be extended to recogni ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
this paper we present a linear time algorithm for the problem of determining if a given dag has a directed leveledplanar embedding. Our algorithm uses a variation of the PQtree data structure introduced by Booth and Lueker [2]. One motivation for our algorithm is that it can be extended to recognize 1queue dags, thus answering an open question in [6]. Combinatorial and algorithmic results related to queue layouts of dags and posets can be found in [4, 7, 5]. Our algorithms also contrasts leveledplanar undirected graphs and leveledplanar dags, since the problem of recognizing leveledplanar graphs has been shown to be NPcomplete by Heath and Rosenberg [8]. Another motivation comes from the importance of the above problem in the area of graph drawing. Our result extends the work of Di Battista and Nardelli [1], Chandramouli and Diwan [3], and Hutton and Lubiw [9]. These authors assume solve the problem assuming certain restrictions on the given dag and leave the general problem open. The organization of the rest of the paper is as follows. Section 2 discusses the nature of the problem and outlines our approach. Section 3 defines the data
Pitfalls of using PQTrees in Automatic Graph Drawing
, 1997
"... A number of erroneous attempts involving PQtrees 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 particula ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
A number of erroneous attempts involving PQtrees 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 PQtree 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.
Minimum level nonplanar patterns for trees
 Proc. Graph Drawing, GD 2007, volume 4875 of LNCS
, 2007
"... Abstract. We add two minimum level nonplanar (MLNP) patterns for trees to the previous set of tree patterns given by Healy et al. [3]. Neither of these patterns match any of the previous patterns. We show that this new set of patterns completely characterize level planar trees. 1 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. We add two minimum level nonplanar (MLNP) patterns for trees to the previous set of tree patterns given by Healy et al. [3]. Neither of these patterns match any of the previous patterns. We show that this new set of patterns completely characterize level planar trees. 1
Linear time planarity testing and embedding of strongly connected cyclic level graphs
"... A level graph is a directed acyclic graph with a level assignment for each node. Such graphs play a prominent role in graph drawing. They express strict dependencies and occur in many areas, e.g., in scheduling problems and program inheritance structures. In this paper we extend level graphs to cyc ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
A level graph is a directed acyclic graph with a level assignment for each node. Such graphs play a prominent role in graph drawing. They express strict dependencies and occur in many areas, e.g., in scheduling problems and program inheritance structures. In this paper we extend level graphs to cyclic level graphs. Such graphs occur as repeating processes in cyclic scheduling, visual data mining, life sciences, and VLSI. We provide a complete study of strongly connected cyclic level graphs. In particular, we present a linear time algorithm for the planarity testing and embedding problem, and we characterize forbidden subgraphs. Our results generalize earlier work on level graphs.