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A linear algorithm for finding a maximal planar subgraph
 SIAM J. Disc. Math
, 2006
"... Abstract. We construct an optimal lineartime algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing ..."
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Abstract. We construct an optimal lineartime algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new optimal planarity testing algorithm. Key words. Planar graphs, planarity testing, incremental algorithms, graph planarization, data structures, triconnectivity. AMS subject classifications. 05C10, 05C85, 68R10, 68Q25, 68W40 1. Introduction. Agraphisplanar
An Analysis of Heuristics for Graph Planarization
 Journal of Information & Optimization Sciences
, 1997
"... We analyze several heuristics for graph planarization, i.e., deleting the minimum number of edges from a nonplanar graph to make it planar. The problem is NPhard, although some heuristics which perform well in practice have been reported. In particular, we compare the two principle methods, based o ..."
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We analyze several heuristics for graph planarization, i.e., deleting the minimum number of edges from a nonplanar graph to make it planar. The problem is NPhard, although some heuristics which perform well in practice have been reported. In particular, we compare the two principle methods, based on path addition and vertex addition, respectively, with a selective edge addition method, an incremental method, and a "cycle packing" approach. For the incremental, the path addition, and the edge addition methods, we prove theoretical worstcase performance bounds of 1=3. We also present an empirical analysis of the heuristics. Our results indicate that the "cyclepacking" method consistently yields the best solutions when applied to a large set of test graphs. 1
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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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.
Angewandte Mathematik und Informatik Universit at zu K oln
, 1998
"... The problem of computing a maximal planar subgraph of a non planar 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. [10]. In this paper we show that the ..."
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The problem of computing a maximal planar subgraph of a non planar 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. [10]. 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 we note that the same holds for a modified version of the algorithm presented by Kant [12]. Our conclusions most likely suggest not to use PQtrees at all for this specific problem. 1 Introduction The minimum number of layers needed in the layout of printed circuit boards and integrated chips is equal to the thickness of the interconnection graph [15]. The thickness of a graph G is the minimum number of planar subgraphs whose union is G. In VLSI design the thickness problem is approximated by successively subtracting large planar subgraphs from a given nonplana...
Fortran Subroutines For Approximate Solution Of Graph Planarization Problems Using Grasp
, 1997
"... . We describe Fortran subroutines for finding approximate solutions of the maximum planar subgraph problem (graph planarization) using a Greedy Randomized Adaptive Search Procedure (GRASP). The design and implementation of the code are described in detail. Computational results with the subroutines ..."
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. We describe Fortran subroutines for finding approximate solutions of the maximum planar subgraph problem (graph planarization) using a Greedy Randomized Adaptive Search Procedure (GRASP). The design and implementation of the code are described in detail. Computational results with the subroutines illustrate the quality of solutions found as a function of number of GRASP iterations. Source code of the FORTRAN subroutines can be downloaded from M.G.C. Resende's Home Page. The URL is http://www.research.att.com/~mgcr/src/gmpsg.tar.gz. Key words. Combinatorial optimization, graph planarization, automatic graph drawing, local search, GRASP, FORTRAN subroutines 1. Introduction. A graph is said to be planar if it can be drawn on the plane in such a way that no two of its edges cross. Given a graph G = (V, E) with vertex set V and edge set E, the objective of graph planarization is to find a minimum cardinality subset of edges F # E such that the graph G # = (V, E \F ) resulting from th...
BranchandBound Techniques for the Maximum Planar Subgraph Problem
, 1994
"... We present branchandbound algorithms for finding a maximum planar subgraph of a nonplanar graph. The problem has important applications in circuit layout, automated graph drawing, and facility layout. The algorithms described utilize heuristics to obtain an initial lower bound for the size of a ..."
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We present branchandbound algorithms for finding a maximum planar subgraph of a nonplanar graph. The problem has important applications in circuit layout, automated graph drawing, and facility layout. The algorithms described utilize heuristics to obtain an initial lower bound for the size of a maximum planar subgraph, then apply a sequence of fast preliminary tests for planarity to eliminate infeasible partial solutions. Computational experience is reported from testing the algorithms on a set of random nonplanar graphs and is encouraging. A bestfirst search technique is shown to be a practical approach to solving problems of moderate size. KEY WORDS: Maximum planar subgraph, branchandbound, planar graph, graph planarization, NPhard, planarity. C.R. CATEGORIES: F.2.2, G.2.2. 1
The LeftRight Planarity Test
, 2009
"... A graph is planar if and only if it can be embedded in the plane without crossings. I give a detailed exposition of simple and efficient, yet poorly known algorithms for planarity testing, embedding, and Kuratowski subgraph extraction based on the leftright characterization of planarity. ..."
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A graph is planar if and only if it can be embedded in the plane without crossings. I give a detailed exposition of simple and efficient, yet poorly known algorithms for planarity testing, embedding, and Kuratowski subgraph extraction based on the leftright characterization of planarity.
PQtrees and maximal planarization  An approach to skewness
, 1998
"... This article formed the basis for our search for a new approach to a skewness heuristic. Given a non planar graph G, then a subgraph G ..."
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This article formed the basis for our search for a new approach to a skewness heuristic. Given a non planar graph G, then a subgraph G