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12
Embedding graphs in books: a layout problem with applications to VLSI design
 SIAM J. ALGEBRAIC DISCRETE METHODS
, 1987
"... We study the graphtheoretic problem of embedding a graph in a book with its vertices in a line along the spine of the book and its edges on the pages in such a way that edges residing on the same page do not cross. This problem abstracts layout problems arising in the routing of multilayer printed ..."
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Cited by 48 (0 self)
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We study the graphtheoretic problem of embedding a graph in a book with its vertices in a line along the spine of the book and its edges on the pages in such a way that edges residing on the same page do not cross. This problem abstracts layout problems arising in the routing of multilayer printed circuit boards and in the design of faulttolerant processor arrays. In devising an embedding, one strives to minimize both the number of pages used and the "cutwidth" of the edges on each page. Our main results (1) present optimal embeddings of a variety of families of graphs; (2) exhibit situations where one can achieve small pagenumber only at the expense of large cutwidth; and (3) establish bounds on the minimum pagenumber of a graph based on various structural properties of the graph. Notable in the last category are proofs that (a) every nvertex dvalent graph can be embedded using O(dn1/2) pages, and (b) for every d>2 and all large n, there are nvertex dvalent graphs whose pagenumber is at least log n]"
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...
Constructing Small Sets That Are Uniform in Arithmetic Progressions
, 2002
"... this paper also satisfy (A ;N ) ..."
The Power of the Queue
 M B
, 1992
"... Queues, stacks, and tapes are basic concepts which have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or lastinfirstout storage) have been thoroughly investigated and are well understood, this is much less the case for queues (firstin ..."
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Cited by 6 (0 self)
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Queues, stacks, and tapes are basic concepts which have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or lastinfirstout storage) have been thoroughly investigated and are well understood, this is much less the case for queues (firstinfirst out storage). In this paper we present a comprehensive study comparing queues to stacks and tapes (offline and with oneway input). The techniques we use rely on Kolmogorov complexity. In particular, 1 queue and 1 tape (or stack) are not comparable: (1) Simulating 1 stack (and hence 1 tape) by 1 queue requires\Omega\Gamma n 4=3 = log n) time in both the deterministic and the nondeterministic cases. (2) Simulating 1 queue by 1 tape requires\Omega\Gamma n 2 ) time in the deterministic case, and\Omega\Gamma n 4=3 =(log n) 2=3 ) in the nondeterministic case; We further compare the relative power between different numbers of queues: (3) Nondeterministically simulating 2 queues...
On Separators, Segregators and Time versus Space
"... We give the first extension of the result due to Paul, Pippenger, Szemeredi and Trotter [24] that deterministic linear time is distinct from nondeterministic linear time. We show that N T IM E(n ..."
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Cited by 6 (0 self)
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We give the first extension of the result due to Paul, Pippenger, Szemeredi and Trotter [24] that deterministic linear time is distinct from nondeterministic linear time. We show that N T IM E(n
Fixed linear crossing minimization by reduction to the maximum cut problem
 in Proc 12th Ann. Int. Computing and Combinatorics Conference (COCOON’06
"... Abstract. Many reallife scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear ..."
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Abstract. Many reallife scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the socalled fixed linear crossing number problem (FLCNP). We show that this N Phard problem can be reduced to the wellknown maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branchandcut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branchandbound algorithms. 1
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 4 (2 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.
Degree constrained book embeddings
 J. Algorithms
, 2002
"... A book embedding of a graph consists of a linear ordering of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph G = (V, ..."
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Cited by 2 (1 self)
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A book embedding of a graph consists of a linear ordering of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph G = (V, E), letf: V → N be a function such that 1 � f(v)�deg(v). We present a Las Vegas algorithm which produces a book embedding of G with O ( � E·maxv⌈deg(v)/f (v) ⌉ ) pages, such that at most f(v)edges incident to avertexvareonasingle page. This result generalises that of Malitz [J. Algorithms 17 (1)
Characterizations 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 th ..."
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Cited by 2 (1 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 >
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.