Results 1  10
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12
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 27 (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 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...
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 13 (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.
The Thickness of a MinorExcluded Class of Graphs
 DISCRETE MATH
, 1998
"... The thickness problem on graphs is NPhard and only few results concerning this graph invariant are known. Using a decomposition theorem of Truemper, we show that the thickness of the class of graphs without G 12  minors is less than or equal to two (and therefore, the same is true for the more wel ..."
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Cited by 4 (0 self)
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The thickness problem on graphs is NPhard and only few results concerning this graph invariant are known. Using a decomposition theorem of Truemper, we show that the thickness of the class of graphs without G 12  minors is less than or equal to two (and therefore, the same is true for the more wellknown class of the graphs without K 5 minors). Consequently, the thickness of this class of graphs can be determined with a planarity testing algorithm in linear time.
Geometric Thickness in a Grid of Linear Area
 In [1
, 2001
"... this paper we consider a variation of geometric thickness which lies between thickness and geometric thickness in which each edge has at most one bend. We are also interested in drawings with small area, which is an important consideration in VLSI and visualisation. To measure the area of a drawing ..."
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Cited by 4 (1 self)
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this paper we consider a variation of geometric thickness which lies between thickness and geometric thickness in which each edge has at most one bend. We are also interested in drawings with small area, which is an important consideration in VLSI and visualisation. To measure the area of a drawing we assume a vertex resolution rule; that is, pairs of vertices are at least unitdistance apart. A drawing obtained from a book embedding by positioning the vertices around a circle, as discussed above, has O(n ) area. The construction in [DEH00] demonstrating that (Kn ) d 4 e has O(n 6 ) area [D. Eppstein, personal communication ]. We prove the following 2dimensional generalisation of the abovementioned result in [Mal94b] for producing book embeddings
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).
Apptopinv  user’s guide
, 2003
"... The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NPcomplete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on HopcroftTarjan planarity testing algorit ..."
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Cited by 1 (1 self)
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The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NPcomplete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on HopcroftTarjan planarity testing algorithm, the spanningtree heuristic and various algorithms based on the cactustree heuristic. Apptopinv contains also a simulated annealing algorithm that can be used to improve the solutions obtained from other heuristics. Most of the heuristics have also a greedy version. We have implemented graph generators for complete graphs, complete kpartite graphs, complete hypercubes, random graphs, random maximum planar and outerplanar graphs and random regular graphs. Apptopinv supports three different graph file formats. Apptopinv is written in C++ programming language for Linuxplatform and GCC 2.95.3 compiler. To compile the program, a commercial LEDA algorithm
A Genetic Algorithm For Determining The Thickness Of A Graph
, 2000
"... The thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed. Determining the thickness of a given graph is known to be a NPcomplete problem. This paper discusses the possibility of determining the thickness of a graph by a genetic algorithm. Our tests s ..."
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Cited by 1 (1 self)
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The thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed. Determining the thickness of a given graph is known to be a NPcomplete problem. This paper discusses the possibility of determining the thickness of a graph by a genetic algorithm. Our tests show that the genetic approach outperforms the earlier heuristic algorithms reported in the literature.