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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...
An InformationTheoretic Upper Bound of Planar Graphs Using Triangulation
, 2003
"... We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n ..."
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Cited by 24 (5 self)
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We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled nnode planar graphs have at least 1.70n edges and at most 2.54n edges.
Planar graphs, via wellorderly maps and trees
 In 30 th International Workshop, Graph  Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science
, 2004
"... Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequ ..."
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Cited by 20 (4 self)
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Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge. 1
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 19 (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 ...
Graph Treewidth and Geometric Thickness Parameters
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
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Cited by 14 (8 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and stararboricity.
Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees
 In: 8 th Annual International Computing & Combinatorics Conference (COCOON). Volume 2387 of LNCS
, 2002
"... We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G. ..."
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Cited by 12 (3 self)
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We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Bijections for Baxter Families and Related Objects
, 2008
"... The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2) ..."
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Cited by 7 (5 self)
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The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)
Stack and Queue Layouts of Halin Graphs
 UNPUBLISHED
, 1995
"... A Halin graph the union of a tree with no degree2 vertices and a cycle on the leaves of the tree. This paper examines the problem of laying out Halin graphs using stacks and queues. A kstack (kqueue) layout of a graph consists of a linear ordering of the vertices along with an assignment of each ..."
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Cited by 7 (0 self)
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A Halin graph the union of a tree with no degree2 vertices and a cycle on the leaves of the tree. This paper examines the problem of laying out Halin graphs using stacks and queues. A kstack (kqueue) layout of a graph consists of a linear ordering of the vertices along with an assignment of each edge to one of k stacks (queues). The ordering and the edge assignments must be made such that if the ordering is traversed from left to right, then each edge can be placed in its assigned stack (queue) when its left endpoint is encountered and removed from its assigned stack (queue) when its right endpoint is encountered. In this paper