Results 1  10
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15
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 29 (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 19 (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 17 (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 14 (7 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 Heuristics for Determining the Thickness of a Graph
 Information Sciences
, 1995
"... We present an empirical analysis of some heuristics for the graph thickness problem, i.e., decomposing a graph into the minimum number of planar subgraphs. The problem has applications in database systems, e.g., the layout of ER diagrams. The heuristics are based on some algorithms for finding a ma ..."
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Cited by 8 (0 self)
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We present an empirical analysis of some heuristics for the graph thickness problem, i.e., decomposing a graph into the minimum number of planar subgraphs. The problem has applications in database systems, e.g., the layout of ER diagrams. The heuristics are based on some algorithms for finding a maximal planar subgraph of a nonplanar graph. Empirical results show that a randomized heuristic is a practical tool for solving this problem. We also improve upon a previously derived upper bound for the thickness of a graph with respect to its size.
Geometric Thickness in a Grid
 Discrete Mathematics
, 2001
"... The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straightline segments, and with each edge assigned to a layer so that no two edges on the same layer cross. We consider a variation on this theme in which each edge is allo ..."
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Cited by 2 (0 self)
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The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straightline segments, and with each edge assigned to a layer so that no two edges on the same layer cross. We consider a variation on this theme in which each edge is allowed one bend. We prove that the vertices of an nvertex medge graph can be positioned in a $$ grid and the edges assigned to $$ layers, so that each edge is drawn with at most one bend and no two edges on the same layer cross. The proof is a 2dimensional generalization of a theorem of S. M. Malitz [J. Algorithms 17(1):7184, 1994] on book embeddings. We obtain a Las Vegas algorithm to compute the drawing in O(m log n log log n) time (with high probability).
Two New Approximation Algorithms for the Maximum Planar Subgraph Problem
, 2006
"... The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NPhard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269302 (1998)] presented the first approximation a ..."
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Cited by 1 (0 self)
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The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NPhard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269302 (1998)] presented the first approximation algorithms for MPS with nontrivial performance ratios. Two algorithms were given, a simple algorithm which runs in linear time for boundeddegree graphs with a ratio 7/18 and a more complicated algorithm with a ratio 4/9. Both algorithms produce outerplanar subgraphs. In this article we present two new versions of the simpler algorithm. The first new algorithm still runs in the same time, produces outerplanar subgraphs, has at least the same performance ratio as the original algorithm, but in practice it finds larger planar subgraphs than the original algorithm. The second new algorithm has similar properties to the first algorithm, but it produces only planar subgraphs. We conjecture that the performance ratios of our algorithms are at least 4/9 for MPS. We experimentally compare the new algorithms against the original simple algorithm. We also apply the new algorithms for approximating the thickness and outerthickness of a graph. Experiments show that the new algorithms produce clearly better approximations than the original simple algorithm by Călinescu et al.