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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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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 efficient implementation of the PCtrees algorithm of shih and hsu’s planarity test
 Institute of Information Science, Academia Sinica
, 2003
"... In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not p ..."
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In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not planar, and how to obtain the embedding alongside the testing algorithm. We have implemented the algorithm using LEDA and an object code is available at
Planarity Algorithms via PQTrees
, 2008
"... We give a lineartime planarity test that unifies and simplifies the algorithms of Shih and Hsu and Boyer and Myrvold; in our view, these algorithms are really one algorithm with different implementations. This leads to a short and direct proof of correctness without the use of Kuratowski’s theorem. ..."
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We give a lineartime planarity test that unifies and simplifies the algorithms of Shih and Hsu and Boyer and Myrvold; in our view, these algorithms are really one algorithm with different implementations. This leads to a short and direct proof of correctness without the use of Kuratowski’s theorem. Our planarity test extends to give a uniform random embedding, to count embeddings, to represent all embeddings, and to give a Kuratowski subgraph of a nonplanar graph. Our algorithm keeps track of possible circular edge orderings in a partial embedding by using a reinterpretation of Booth and Lueker’s PQtree data structure. This is a classic data structure that represents certain sets of permutation and gives lineartime algorithms for various matrix and graph ordering problems. We show that our reinterpretation of PQtrees gives exactly the PCtrees of Shih and Hsu. We give a simpler and more symmetric implementation of PQtree reduction. This simplifies various applications and leads to an efficient algorithm for a generalization of the consecutive and circular ones problems. 1
Planarity Testing and Embedding
, 2004
"... Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 19 ..."
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Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 1930, an efficient solution to it was found only in the seventies of the last century. Planar graphs play an important role both in the graph theory and in the graph drawing areas. In fact, planar graphs have several interesting properties: for example they are sparse, fourcolorable, allow a number of operations to be performed efficiently, and their structure can be elegantly described by an SPQRtree (see Section 3.1.2). From the information visualization perspective, instead, as edge crossings turn out to be the main culprit for reducing readability, planar drawings of graphs are considered clear and comprehensible. As a matter of fact, the study of planarity has motivated much of the development of graph theory. In this chapter we review the number of alternative algorithms available in the literature for efficiently testing planarity and computing planar embeddings. Some of these algorithms