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20
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.
Geometric Thickness of Complete Graphs
 J. GRAPH ALGORITHMS APPL
, 2000
"... We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straightline edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantiti ..."
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Cited by 32 (4 self)
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We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straightline edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graphtheoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, {Kn}. We show that the geometric thickness of Kn lies between #(n/5.646) + 0.342# and #n/4#, and we give exact values of the geometric thickness of Kn for n # 12 and n #{15, 16}. We also consider the geometric thickness of the family of complete bipartite graphs. In particular, we show that, unlike the case of complete graphs, there are complete bipartite graphs with arbitrarily large numbers of vertices for which the geometric thickness coincides with the standard graphtheoretical thickness.
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 21 (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...
Separating Thickness from Geometric Thickness
, 2002
"... We show that graphtheoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness ≥ t. ..."
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Cited by 19 (2 self)
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We show that graphtheoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness ≥ t.
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 18 (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.
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
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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Cited by 11 (0 self)
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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.
The Visibility Number Of A Graph
 SIAM J. DISCRETE MATH
, 1998
"... We introduce the visibility number b(G) of a graph G, which is the minimum t such that G can be represented by assigning each vertex a union of at most t horizontal segments in the plane so that vertices u; v are adjacent if and only if some point assigned to u sees some point assigned to v via a ve ..."
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Cited by 4 (1 self)
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We introduce the visibility number b(G) of a graph G, which is the minimum t such that G can be represented by assigning each vertex a union of at most t horizontal segments in the plane so that vertices u; v are adjacent if and only if some point assigned to u sees some point assigned to v via a vertical segment unobstructed by other assigned points. We prove the following: 1) every planar graph has visibility number at most 2, which is sharp. 2) r b(K m;n ) r + 1, where r = d(mn + 4)=(2m + 2n)e. 3) dn=6e b(K n ) dn=6e + 1. 4) When G has n vertices, b(G) dn=6e + 2.
Three Drexel Information Science Research Studies
, 1967
"... AND ITS APPLICATION TO BIOMEDICAL LITERATURE F. Lunin . . . . . . 47 I ..."
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AND ITS APPLICATION TO BIOMEDICAL LITERATURE F. Lunin . . . . . . 47 I