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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...
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 16 (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
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
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.
Bar kVisibility Graphs
"... Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a onetoone correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if t ..."
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Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a onetoone correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar kvisibility graphs. We obtain an upper bound on the number of edges in a bar kvisibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1visibility graph. We conjecture that bar 1visibility graphs have thickness at most 2.
unknown title
, 1999
"... Selfcomplementary graphs and generalisations: a comprehensive reference manual ..."
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Selfcomplementary graphs and generalisations: a comprehensive reference manual
unknown title
, 2004
"... Abstract The bar visibility number of a graph G, denoted b(G), is the minimum t such that G can be represented by assigning each vertex x the set Sx of points in at most t horizontal segments in the plane so that uv 2 E(G) if and only if some point of Su sees some point of Sv via a vertical segment ..."
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Abstract The bar visibility number of a graph G, denoted b(G), is the minimum t such that G can be represented by assigning each vertex x the set Sx of points in at most t horizontal segments in the plane so that uv 2 E(G) if and only if some point of Su sees some point of Sv via a vertical segment of positive width unobstructed by assigned points. Among our results are the following: 1) Every planar graph has bar visibility number at most 2, which is sharp. 2) r ^ b(Km;n) ^ r + 1, where r = l mn+4
RECENT RESULTS IN TOPOLOGICAL GRAPH THEORY* By
"... A graph G is usually defined as a finite collection V of points together with a collection X of lines, each of which joins two distinct points and no two of which join the same pair of points. This combinatorial definition asserts nothing about drawing graphs on surfaces such as the plane, sphere, t ..."
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A graph G is usually defined as a finite collection V of points together with a collection X of lines, each of which joins two distinct points and no two of which join the same pair of points. This combinatorial definition asserts nothing about drawing graphs on surfaces such as the plane, sphere, torus, projective plane etc. The purpose of this lecture is to explore some of these topological aspects of graph theory and to describe a few unsolved problems concerning them. In order to fix the terminology of this iecture, we begin by drawing all the graphs with four points: