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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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Cited by 32 (0 self)
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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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Cited by 18 (0 self)
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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...
On the thickness of graphs of given degree
 Inform. Sci
, 1991
"... The results presented here refer to the determination of the thickness of a graph; that is, the minimum number of planar subgraphs into which the graph can be decomposed. A useful general, preliminary result obtained is Theorem 8: that a planar graph always has a planar representation in which the n ..."
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Cited by 15 (0 self)
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The results presented here refer to the determination of the thickness of a graph; that is, the minimum number of planar subgraphs into which the graph can be decomposed. A useful general, preliminary result obtained is Theorem 8: that a planar graph always has a planar representation in which the nodes are placed in arbitrary given positions. It is then proved that, if we have positive integers D and T, such that any graph of degree at most D has thickness at most T: Theorem 9: any graph of degree d has thickness at most T roof { ( d + 1) I D}; Theorem 10: any graph of degree dean always be embedded in a regular graph G 0 of any degree f;. d; Corollary 5: any graph of degree dhas thickness at most roof(d/2); Theorem 12: with D and T defined as above, we have D.;; 4 T 2; Corollary 6: if T = 2, then D.;; 6. We further conjecture that, indeed, the thickness of any graph of degree not exceeding 6 is never more than 2. Since the design and fabrication of VLSI c0111puter chips is essentially a concrete representation of the planar decomposition of a graph, all these results are of direct practical interest. DEDICATION This paper is humbly and affectionately dedicated to my mother, Anne Halton, whose indomitable hope and courageous perseverance in the face of difficulty have been an admirable example to me throughout my life.sine qua non~~ ~ DEFINITIONS Let N = { V1, V2, • o o, vn} be a finite set of nodes (or vertices) and write L(N) = { {x, y}: x E N A y e N A x # y} for the set of all possible edges (i.e., pairs of nodes). If E c; G = (N, E) = (N(G), E(G)) L(N), we call a graph (more precisely, an undirected graph), with n = I Nl nodes specified by
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.
On Graph Thickness, Geometric Thickness, and Separator Theorems
"... We investigate the relationship between geometric thickness and the thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O(log n). The technique used can be extended to other classes of graphs ..."
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We investigate the relationship between geometric thickness and the thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O(log n). The technique used can be extended to other classes of graphs so long as a standard separator theorem exists. For example, we can apply it to show the known bound that thickness two graphs have geometric thickness O ( √ n), yielding a simple construction in the process. 1