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17
Maximum Planar Subgraphs and Nice Embeddings: Practical Layout Tools
 ALGORITHMICA
, 1996
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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...
An Approach for Mixed Upward Planarization
 In Proc. 7th International Workshop on Algorithms and Data Structures (WADS’01
, 2003
"... In this paper, we consider the problem of finding a mixed upward planarization of a mixed graph, i.e., a graph with directed and undirected edges. The problem is a generalization of the planarization problem for undirected graphs and is motivated by several applications in graph drawing. ..."
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Cited by 15 (2 self)
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In this paper, we consider the problem of finding a mixed upward planarization of a mixed graph, i.e., a graph with directed and undirected edges. The problem is a generalization of the planarization problem for undirected graphs and is motivated by several applications in graph drawing.
Randomized Graph Drawing with HeavyDuty Preprocessing
 In: AVI ’94: Proceedings of the Workshop on Advanced Visual Interfaces
, 1994
"... : We present a graph drawing system for general undirected graphs with straightline edges. It carries out a rather complex set of preprocessing steps, designed to produce a topologically good, but not necessarily nicelooking layout, which is then subjected to Davidson and Harel's simulated anneali ..."
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Cited by 14 (1 self)
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: We present a graph drawing system for general undirected graphs with straightline edges. It carries out a rather complex set of preprocessing steps, designed to produce a topologically good, but not necessarily nicelooking layout, which is then subjected to Davidson and Harel's simulated annealing beautification algorithm. The intermediate layout is planar for planar graphs and attempts to come close to planar for nonplanar graphs. The system's results are significantly better, and much faster, than what the annealing approach is able to achieve on its own. 1 Introduction A large amount of work on the problem of graph layout has been carried out in recent years, resulting in a number of sophisticated and powerful algorithms. An extensive and detailed survey can be found in [BETT93]. Many of the approaches taken are limited to special cases of graphs, such as trees or planar graphs; others concentrate on special kinds of layouts, such as rectilinear grid drawings, or convex drawin...
Pitfalls of using PQTrees in Automatic Graph Drawing
, 1997
"... A number of erroneous attempts involving PQtrees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particula ..."
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Cited by 10 (0 self)
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A number of erroneous attempts involving PQtrees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particular, we examine erroneous usage of the PQtree data structure in algorithms for computing maximal planar subgraphs and an algorithm for testing leveled planarity of leveled directed acyclic graphs with several sources and sinks.
Graph Planarization and Skewness
"... The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NPComplete. Several heuristics for the problem have been devised but their worstcase performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heurist ..."
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Cited by 10 (0 self)
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The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NPComplete. Several heuristics for the problem have been devised but their worstcase performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heuristic, based on spanning trees, for generating a subgraph with size at least 2/3 of the optimum for any input graph. The skewness of the ndimensional hypercube Qn is also derived. Finally, we explore the relationship between the skewness and crossing number of a graph.
Solving the Maximum Weight Planar Subgraph Problem by Branch and Cut
 PROCEEDINGS OF THE THIRD CONFERENCE ON INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION
, 1993
"... In this paper we investigate the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is defined and studied. All subgraphs of a graph G, which are subdivisions of K 5 or K 3 ..."
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Cited by 8 (1 self)
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In this paper we investigate the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is defined and studied. All subgraphs of a graph G, which are subdivisions of K 5 or K 3;3 , turn out to define facets of this polytope. We also present computational experience with a branch and cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision of K 5 or K 3;3 . These structures give us inequalities which are used as cutting planes.
A linear time algorithm for finding a maximal planar subgraph based on PCtrees
 of Lecture Notes in Computer Science
, 2005
"... ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorith ..."
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Cited by 5 (0 self)
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ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2]. A recent O(mα (n)) algorithm was obtained by La Poute [7]. Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depthfirstsearch ordering. The planarity test [5] uses no complicated data structure and is conceptually simpler than Hopcroft and Tarjan's path addition and Lempel, Even and Cederbaum's vertex addition approaches. 1 1.
An Analysis of Some Heuristics for the Maximum Planar Subgraph Problem
 Proc. 6 th Annual ACMSIAM Symp. on Discrete Algorithms
, 1995
"... Introduction The problem of extracting a maximum planar subgraph from a nonplanar graph, referred to as graph planarization, has important applications in circuit layout, facility layout, and automated graphical display systems [F, TDB]. The problem is NPhard [LG]; hence, research has focused on h ..."
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Cited by 4 (0 self)
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Introduction The problem of extracting a maximum planar subgraph from a nonplanar graph, referred to as graph planarization, has important applications in circuit layout, facility layout, and automated graphical display systems [F, TDB]. The problem is NPhard [LG]; hence, research has focused on heuristics. There are several algorithms for finding maximal planar subgraphs [CHT, CNS, GT, JTS, JM, K, OT]. However, there are graphs (see [CC]) for which the size ratio between two maximal planar subgraphs can be as small as 1=3. Hence, unless some precautions are taken to avoid the extraction of small subgraphs, these heuristics have the potential for poor behavior. In this paper, we analyze the worstcase performance of some heuristics and show that there are graphs which can cause each of them to achieve the 1=3 bound. However, a theoretical analysis of an algorithm's performance is often too pessimistic and somew
On Computing a Maximal Planar Subgraph using PQTrees
, 1996
"... The problem of computing a maximal planar subgraph of a nonplanar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQtrees. The latest attempt has been reported by Jayakumar et al. (1989). In this paper we show that t ..."
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Cited by 4 (3 self)
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The problem of computing a maximal planar subgraph of a nonplanar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQtrees. The latest attempt has been reported by Jayakumar et al. (1989). In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and that the same holds for a modified version of the algorithm presented by Kant (1992). Our conclusions most likely suggest not to use PQtrees at all for this specific problem.