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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 33 (22 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...
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 ..."
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Cited by 28 (3 self)
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. 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...
Stack And Queue Layouts Of Posets
 SIAM J. Discrete Math
, 1995
"... . The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower ..."
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. The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower bound of \Omega\Gamma p n) is shown for the queuenumber of the class of nelement planar posets. The queuenumber of a planar poset is shown to be within a small constant factor of its width. The stacknumber of nelement posets with planar covering graphs is shown to be \Theta(n). These results exhibit sharp differences between the stacknumber and queuenumber of posets as well as between the stacknumber (queuenumber) of a poset and the stacknumber (queuenumber) of its covering graph. Key words. poset, queue layout, stack layout, book embedding, Hasse diagram, jumpnumber AMS subject classifications. 05C99, 68R10, 94C15 1. Introduction. Stack and queue layouts of undirected graphs appear ...
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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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 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.
Treepartitions of ktrees with applications in graph layout
 Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science (WG’03
, 2002
"... Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result t ..."
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Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result to two wellstudied models of graph layout are presented. First it is proved that graphs of bounded treewidth have bounded queuenumber, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded treewidth have threedimensional straightline grid drawings with linear volume, which represents the largest known class of graphs with such drawings. 1
Scheduling TreeDags Using FIFO Queues: A Controlmemory Tradeoff
"... We study here a combinatorial problem that is motivated by a genre of architectureindependent scheduler for parallel computations. Such schedulers are often used, for instance, when computations are being done by a cooperating network of workstations. The results we obtain expose a controlmemory t ..."
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Cited by 12 (0 self)
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We study here a combinatorial problem that is motivated by a genre of architectureindependent scheduler for parallel computations. Such schedulers are often used, for instance, when computations are being done by a cooperating network of workstations. The results we obtain expose a controlmemory tradeoff for such schedulers, when the computation being scheduled has the structure of a complete binary tree. The combinatorial problem takes the following form. Consider, for each integer N =2 n, a family of n algorithms for linearizing the Nleaf complete binary tree in such away that each nonleaf node precedes its children. For each k 2f1 � 2�:::�ng, the kth algorithm in the family employs k FIFO queues to e ect the linearization, in a manner specified later (cf., [1], [5] [7]). In this paper, we expose a tradeoff between the number of queues used by eachofthen algorithms  which we view as measuring the control complexity of the algorithm  and the memory requirements of the algorithms, as embodied in the required capacity ofthe largestcapacity queue. Specifically, we prove that, for each k 2f1 � 2�:::�ng, the maximum perqueue capacity, call it Q k(N), for a kqueue algorithm that linearizes an Nleaf complete binary tree satisfies e
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 ..."
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Cited by 11 (1 self)
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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
CHARACTERISATIONS AND EXAMPLES OF GRAPH CLASSES WITH BOUNDED EXPANSION
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of t ..."
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Cited by 10 (3 self)
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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several lineartime algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of socalled topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded nonrepetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologicallyclosed class, while graphs with bounded crossing number are contained in a minorclosed class.