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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...
A Simple Linear Time Algorithm for Embedding Maximal Planar Graphs
, 1993
"... All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex datastructures, or they are easy to understand and implement, but require more than linear time [ ..."
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All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex datastructures, or they are easy to understand and implement, but require more than linear time [2]. In this paper, a new lineartime algorithm for embedding maximal planar graphs is proposed. This algorithm is both easy to understand and easy to implement. The algorithm consists of three phases which use only simple, local graphmodifications. In addition to planar embedding, the new algorithm allows to test graphs for maximal planarity. The generation of Straight Line Drawings by a technique of Read [12] can be naturally incorporated into the algorithm. We also demonstrate how to generate random (maximal) planar graphs. The algorithm presented constitutes a first step towards a simple, lineartime solution for embedding general planar graphs. 2 3 1 Introduction One of the main top...
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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Cited by 1 (0 self)
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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
Contractions, Removals and How to Certify 3Connectivity in Linear Time
"... One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the exis ..."
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One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K4, such that every intermediate graph is 3vertexconnected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(V  2) to O(E). This result has a number of consequences; an important one is a new lineartime test of 3connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in the last years. The test is conceptually different from wellknown lineartime 3connectivity tests and uses a certificate that is easy to verify in time O(E). We show how to extend the results to an optimal certifying test of 3edgeconnectivity. 1
Planarity Testing of Graphs 10.1
"... Introduction We are interested in the following problem: given an undirected graph G, determine if G is planar (ican be drawn on a paper without having any two edges crossj). The algorithm developed in this lecture attempts to embed G in the plane. If it succeeds it will output the embedding (at t ..."
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Introduction We are interested in the following problem: given an undirected graph G, determine if G is planar (ican be drawn on a paper without having any two edges crossj). The algorithm developed in this lecture attempts to embed G in the plane. If it succeeds it will output the embedding (at this point we are deliberately vague about exactly what this means) and otherwise it will output ino.j This is done by embedding one path at a time where the endpoints on the path are vertices that have already been embedded. Which path to embed, and how to embed it, is chosen in a greedy fashion. The diOEculty is to show that the greedy choice used will always nd an embedding, assuming one exists. The complexity of the algorithm is O(jV j 2 ). Hopcroft and Tarjan [1] present an algorithm that solves the problem in time O(jV j) based on a p