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On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 36 (23 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
On the Parameterized Complexity of Layered Graph Drawing
 PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01
, 2001
"... We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for ..."
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Cited by 24 (8 self)
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We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a lineartime algorithm to decide if a graph has a crossingfree hlayer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossingfree drawing (for fixed k or r). If the number of crossings or deleted edges is a nonfixed parameter then these problems are NPcomplete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the socalled Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
Level Planar Embedding in Linear Time
, 1999
"... A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be ..."
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Cited by 21 (0 self)
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A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) 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 V i are drawn on the line li  {(x, k  i) ] x ), the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to
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 ..."
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Cited by 18 (8 self)
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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 Approach for Mixed Upward Planarization
 In Proc. 7th International Workshop on Algorithms and Data Structures (WADS’01
, 2003
"... In this paper, we consider the problem of finding a mixed upward planarization of a mixed graph, i.e., a graph with directed and undirected edges. The problem is a generalization of the planarization problem for undirected graphs and is motivated by several applications in graph drawing. ..."
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Cited by 17 (2 self)
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In this paper, we consider the problem of finding a mixed upward planarization of a mixed graph, i.e., a graph with directed and undirected edges. The problem is a generalization of the planarization problem for undirected graphs and is motivated by several applications in graph drawing.
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 ..."
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Cited by 15 (7 self)
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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.
Toward a theory of planarity: Hananitutte and planarity variants
 In Graph Drawing, Lecture Notes in Computer Science
, 2013
"... Abstract. We study HananiTutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for pla ..."
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Abstract. We study HananiTutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested in the writings of Tutte and Wu. 1
A parameterized algorithm for upward planarity testing
 In Annual European Symposium on Algorithms (Proc. ESA ’04
, 2004
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which eac ..."
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Cited by 7 (0 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NPcomplete. Parameterized complexity is a technique used to find efficient algorithms for hard
Minimum level nonplanar patterns for trees
, 2007
"... Abstract. Minimum level nonplanar (MLNP) patterns play the role for level planar graphs that the forbidden Kuratowksi subdivisions K5 and K3,3 play for planar graphs. We add two MLNP patterns for trees to the previous set of tree patterns given by Healy et al. [4]. Neither of these patterns match an ..."
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Abstract. Minimum level nonplanar (MLNP) patterns play the role for level planar graphs that the forbidden Kuratowksi subdivisions K5 and K3,3 play for planar graphs. We add two MLNP patterns for trees to the previous set of tree patterns given by Healy et al. [4]. Neither of these patterns match any of the previous patterns. We show that this new set of patterns completely characterizes level planar trees. 1
The VertexExchange Graph and its use in MultiLevel Graph Layout
 In Kratochvil [Kra99
, 1999
"... In this paper we consider the problems of testing a multilevel graph for planarity and laying out or, drawing, a multilevel graph in a clear way. Our approach is based on integer linear programming. We introduce a new abstraction of a common integer linear programming formulation of the problems t ..."
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In this paper we consider the problems of testing a multilevel graph for planarity and laying out or, drawing, a multilevel graph in a clear way. Our approach is based on integer linear programming. We introduce a new abstraction of a common integer linear programming formulation of the problems that we call a vertexexchange graph. We demonstrate how this concept can be used to solve the problems by providing clear and simple algorithms for testing a multilevel graph for planarity and laying out a multilevel graph when planar. We also show how the concept can be used to derive other equalities and inequalities that are of use when determining a graph layout with the smallest number of edge crossings. We provide results of experiments conducted to test these new constraints. 1 Introduction Multilevel layout is a wellknown graph layout problem. Vertices of a graph are placed on discrete levels and edges are allowed only between vertices of the adjacent levels. Traditionally, mult...