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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 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...
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 26 (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...
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.
Stability, Fairness and Scalability of MultiAgent Systems
 International Journal of KnowledgeBased Intelligent Engineering Systems
, 1999
"... : We study a general network of agents that can be built with Zeus [18]. Relationships between agents can be peer, slave, master, discounter, or no relation at all. There are four possible strategies: the cheapest agent is selected, preference to slaves rst, cutprice discounting based on the util ..."
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Cited by 10 (4 self)
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: We study a general network of agents that can be built with Zeus [18]. Relationships between agents can be peer, slave, master, discounter, or no relation at all. There are four possible strategies: the cheapest agent is selected, preference to slaves rst, cutprice discounting based on the utility, and cheapest agent chosen, but preference given to cheapest slave. The cost of a task for the agent originating it, is the cost of the resources used. The size of the initial endowment is determined so that there are never any lost tasks in the system. We also establish the inuence of agent strategies on stability and fairness. We were able to determine how much discounting the agents can allow, and how to control the coordination time. The growth of the maximum communication load with respect to the number of agents is calculated for various topologies of networks of agents. A performance measure related to the speed of the network is also calculated. Keywords: Intelligent ag...
On the Queue Number of Planar Graphs
, 2010
"... We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1 ..."
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Cited by 3 (0 self)
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We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1
New Results in Graph Layout
 School of Computer Science, Carleton Univ
, 2003
"... A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models o ..."
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Cited by 1 (1 self)
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A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give analogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta [Proc. Graph Drawing, 2001] and Pach, Thiele, and Toth [Proc. Graph Drawing, 1997] concerning threedimensional straightline grid drawings. We initiate the study of threedimensional polyline grid drawings and establish volume bounds within a logarithmic factor of optimal.
Boundeddegree graphs have arbitrarily large geometric thickness
 Electron. J. Combin
, 509
"... It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c. ..."
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Cited by 1 (1 self)
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It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c.
1 Graph Layouts via Layered Separators
"... Abstract. A kqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of ..."
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Abstract. A kqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queuenumber (tracknumber) of a graph G, is the minimum k such that G has a kqueue (ktrack) layout. This paper proves that every nvertex planar graph has track number and queue number at most O(logn). This improves the result of Di Battista, Frati and Pach [Foundations of Computer Science, (FOCS ’10), pp. 365–374] who proved the first subpolynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log 2 n) queue number and O(log 8 n) track number bounds for planar graphs. The result also implies that every planar graph has a 3D crossingfree grid drawing in O(nlogn) volume. The proof uses a nonstandard type of graph separators. 1
Queue layouts of hypercubes ∗
, 2011
"... A queue layout of a graph consists of a linear ordering σ of its vertices, and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to σ. We show that the ndimensional hypercube Qn has a layout into n−⌊log2 n⌋ queues for all n ≥ 1. On the oth ..."
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A queue layout of a graph consists of a linear ordering σ of its vertices, and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to σ. We show that the ndimensional hypercube Qn has a layout into n−⌊log2 n⌋ queues for all n ≥ 1. On the other hand, for every ε> 0 every queue layout of Qn has more than ( 1 2 − ε)n − O(1/ε) queues, and in particular, more than (n − 2)/3 queues. This improves previously known upper and lower bounds on the minimal number of queues in a queue layout of Qn. For the lower bound we employ a new technique of outin representations and contractions which may be of independent interest.