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19
StraightLine Drawings on Restricted Integer Grids in Two and Three Dimensions (Extended Abstract)
, 2002
"... This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straightline crossingfree drawings with vertices located at the grid points of phi? We characterize the trees t ..."
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Cited by 38 (4 self)
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This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straightline crossingfree drawings with vertices located at the grid points of phi? We characterize the trees that can be drawn on a two dimensional c * n × k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossingfree with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal  it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossingfree straight line 3d drawings in linear volume for a nontrivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.
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 31 (19 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...
Layout of Graphs with Bounded TreeWidth
 2002, submitted. Stacks, Queues and Tracks: Layouts of Graph Subdivisions 41
, 2004
"... A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a gr ..."
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Cited by 26 (20 self)
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A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a graph represents the vertices by points in Z and the edges by noncrossing linesegments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of threedimensional drawing of a graph G is closely related to the queuenumber of G. In particular, if G is an nvertex member of a proper minorclosed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queuenumber.
Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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Cited by 24 (12 self)
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
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 19 (9 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.
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 13 (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.
A FixedParameter Approach to TwoLayer Planarization
, 2002
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar ..."
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Cited by 12 (4 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NPcomplete, as is the 1Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +G) algorithm for 2Layer Planarization and an O(3 k ·G) algorithm for 1Layer Planarization, thus achieving linear time for fixed k.
An Efficient Fixed Parameter Tractable Algorithm for 1Sided Crossing Minimization
 ALGORITHMICA
, 2004
"... We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings. ..."
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Cited by 11 (4 self)
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We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 9 (1 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
Fixed parameter algorithms for onesided crossing minimization revisited
 Graph Drawing
, 2004
"... We exhibit a small problem kernel for the onesided crossing minimization problem. This problem plays an important role in graph drawing algorithms based on the Sugiyama layering approach. Moreover, we improve on the search tree algorithm developed in [7] and derive an O(1.4656 k + kn 2) algorithm f ..."
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Cited by 9 (3 self)
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We exhibit a small problem kernel for the onesided crossing minimization problem. This problem plays an important role in graph drawing algorithms based on the Sugiyama layering approach. Moreover, we improve on the search tree algorithm developed in [7] and derive an O(1.4656 k + kn 2) algorithm for this problem, where k upperbounds the number of tolerated edge crossings in the drawings of an nvertex graph.