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Book embeddings of graphs and a theorem of Whitney
, 2003
"... It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a twopage book embedding. The latter extends a theorem of H. Whitney and ..."
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It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a twopage book embedding. The latter extends a theorem of H. Whitney and gives twopage book embeddings for Xtrees and square grids.
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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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.
Figures of merit: The sequel
"... Science has marched on despite the appearance of the original "Figures of Merit" [18]. The purpose of this survey is to bring the community up to date on the most recent bounds, so that we may collaborate to improve them. ..."
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Science has marched on despite the appearance of the original "Figures of Merit" [18]. The purpose of this survey is to bring the community up to date on the most recent bounds, so that we may collaborate to improve them.
Stacks, Queues and Tracks: Layouts of Graph Subdivisions
, 2005
"... A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of ea ..."
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A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges 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 stacknumber (respectively, queuenumber, tracknumber) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a kstack (kqueue, ktrack) layout. This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3stack subdivision. The best known upper bound on the number of division vertices per edge in a 3stack subdivision of an nvertex graph G is improved from O(log n) to O(log min{sn(G), qn(G)}). This result reduces the question of whether queuenumber is bounded by stacknumber to whether 3stack graphs have bounded queue number. It is proved that every graph has a 2queue subdivision, a 4track subdivision, and a mixed 1stack 1queue subdivision. All these values are optimal for every nonplanar graph. In addition, we characterise those graphs with kstack, kqueue, and ktrack subdivisions, for all values of k. The number of division vertices per edge in the case of 2queue and 4track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we
The Pagenumber of Genus g Graphs is 0(g)
, 1992
"... In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g> 1 have unbounded pagenumber. In this paper. it is proven that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An 0 ( fi) lower bound is also derwed. The first algorithm in the literature for embe ..."
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In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g> 1 have unbounded pagenumber. In this paper. it is proven that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An 0 ( fi) lower bound is also derwed. The first algorithm in the literature for embedding an arbitra ~ graph in a book with a nontrlwal upper bound on the number of pages M presented. First, the algorithm computes the genus g of a graph using the algorithm of Filotti, Miller, Reif ( 1979), which is polynomialtime for fixed genus. Second, it applies an optimaltime algorithm for obtaining an 0 ( g)page book embedding. Separate book embedding algorithms are given for the cases of graphs embedded m orlentable and nonorientable surfaces. An important aspect of the construction is a new decomposition theorem, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results obtained: faulttolerant VLSI and complexity theory,