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Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
The Thickness of Graphs: A Survey
 Graphs Combin
, 1998
"... We give a stateoftheart survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a ba ..."
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We give a stateoftheart survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a decomposition of a graph in planar subgraphs. Key words: Thickness, maximum planar subgraph, branch and cut 1 Introduction In VLSI circuit design, a chip is represented as a hypergraph consisting of nodes corresponding to macrocells and of hyperedges corresponding to the nets connecting the cells. A chipdesigner has to place the macrocells on a printed circuit board (which usually consists of superimposed layers), according to several designing rules. One of these requirements is to avoid crossings, since crossings lead to undesirable signals. It is therefore desirable to find ways to handle wi...
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 20 (9 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.
Relations among embedding parameters for graphs
 In Graph theory, combinatorics, and applications
, 1991
"... Let G be a simple graph with nvertices, and let aj, bt, y; and e denote, respectively, the arboricity, book thickness '(also called pagenwnber), genus, and thickness of G. We establish inequalities, each ofwhich is best possible up to c:z constant, between pairs ofthese parameters. In particula ..."
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Let G be a simple graph with nvertices, and let aj, bt, y; and e denote, respectively, the arboricity, book thickness '(also called pagenwnber), genus, and thickness of G. We establish inequalities, each ofwhich is best possible up to c:z constant, between pairs ofthese parameters. In particular, we show: e~6+~2r2, aj ~ bt + 1, and r ~ (81) (n1). 1.
Biplanar crossing numbers I: A Survey of Results and Problems
 IN: MORE SETS, GRAPHS AND NUMBERS
, 2006
"... We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We find a few particular exact values and give general lower and upper bounds for the biplanar crossing n ..."
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We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We find a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We find the exact biplanar crossing number of K 5;q for every q.
Topological Graph Theory from Japan
"... This is a survey of studies on topological graph theory developed by Japanese people in the recent two decades and presents a big bibliography including almost all papers written by Japanese topological graph theorists. ..."
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This is a survey of studies on topological graph theory developed by Japanese people in the recent two decades and presents a big bibliography including almost all papers written by Japanese topological graph theorists.