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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...
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 32 (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...
ThreeDimensional Grid Drawings with SubQuadratic Volume
, 1999
"... A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight linesegments representing the edges are pairwise noncrossing. A O(n volume bound is proved for threedimensional grid drawings of graphs with bounded ..."
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Cited by 21 (15 self)
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A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight linesegments representing the edges are pairwise noncrossing. A O(n volume bound is proved for threedimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n ). These results (partially) solve open problems due to Pach, Thiele, and Toth (1997) and Felsner, Liotta, and Wismath (2001).
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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Cited by 21 (4 self)
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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 ...
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
Queue layouts, treewidth, and threedimensional graph drawing
 Proc. 22nd Foundations of Software Technology and Theoretical Computer Science (FST TCS '02
, 2002
"... Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., ..."
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Cited by 10 (7 self)
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Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every nvertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing onedimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queuenumber. Let G be an nvertex member of a proper minorclosed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queuenumber. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queuenumber? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded treewidth have bounded queuenumber? We prove that graphs with bounded pathwidth, or both bounded treewidth and bounded maximum degree, have bounded queuenumber. As a corollary we obtain threedimensional drawings with optimal O(n) volume, for seriesparallel graphs, and graphs with both bounded treewidth and bounded maximum degree. 1
On Crossing Sets, Disjoint Sets and the Pagenumber
, 1998
"... Let G = (V; E) be a tpartite graph with n vertices and m edges, where the partite sets are given. We present an O(n 2 m 1:5 ) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set ..."
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Cited by 9 (0 self)
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Let G = (V; E) be a tpartite graph with n vertices and m edges, where the partite sets are given. We present an O(n 2 m 1:5 ) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set of disjoint (pairwise noncrossing) edges are O( p t \Delta m). As an application we embed G in a book of O( p t \Delta m) pages, in O(n 2 m 1:5 ) time, resolving an open question for the pagenumber problem. A similar result is obtained for the dual of the pagenumber or the queuenumber. Our algorithms are obtained by derandomizing a probabilistic proof. 1 Introduction and Summary 1.1 Preliminaries Throughout this paper G = (V; E) is an undirected graph with jV j = n and jEj = m. A linear ordering of a set S is a bijection from S to f1; 2; : : : ; jSjg. Let h be a linear ordering of V . Consider a drawing of G that is obtained by placing the vertices along a straight line in the pl...
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 9 (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.