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On the cutting edge: Simplified O(n) planarity by edge addition
 Journal of Graph Algorithms and Applications
, 2004
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Stop Minding Your P's and Q's: A Simplified O(n) Planar Embedding Algorithm
 In Proc. 10th ACMSIAM Symposium on Discrete Algorithms, SODA
, 1999
"... A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but all are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the comp ..."
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Cited by 25 (4 self)
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A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but all are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the complexities of previous approaches (including the need to first stnumber the vertices). Our new algorithm easily permits the extraction of a planar obstruction (a subgraph homeomorphic to K3;3 or K5) in O(n) time if the graph is not planar. Our algorithm is similar to the algorithm of Booth and Lueker which uses a data structure called a PQtree. The Pnodes in a PQtree represent parts of the partially embedded graph that can be permuted, and the Qnodes represent parts that can be flipped. We avoid the use of Pnodes by not connecting pieces together until they become biconnected. We avoid Q nodes by using a data structure which allows biconnected components to be flipped in O(1) time. 1 In...
Efficient Extraction of Multiple Kuratowski Subdivisions (TR)
, 2007
"... Abstract. A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for nonplanarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is ..."
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Cited by 2 (1 self)
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Abstract. A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for nonplanarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., BranchandCut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation. 1
On CotreeCritical and DFS CotreeCritical Graphs
, 2003
"... We give a characterization of DFS cotreecritical graphs which is central to the linear time Kuratowski finding algorithm implemented in PIGALE (Public Implementation of a Graph Algorithm Library and Editor [2]) by the authors, and deduce a justification of a very simple algorithm for finding a Kura ..."
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Cited by 1 (0 self)
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We give a characterization of DFS cotreecritical graphs which is central to the linear time Kuratowski finding algorithm implemented in PIGALE (Public Implementation of a Graph Algorithm Library and Editor [2]) by the authors, and deduce a justification of a very simple algorithm for finding a Kuratowski subdivision in a DFS cotreecritical graph.
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
Another characterisation of planar graphs
"... A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph. 1 ..."
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A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph. 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.
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
Simplified O(n) Planarity Algorithms
, 2001
"... A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This pape ..."
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A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This paper presents an O(n) planar embedding algorithm that avoids a number of the complexities of prior approaches (an early version of this work was presented at the January 1999 Symposium on Discrete Algorithms).