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PC trees and circularones arrangements
 Theoretical Computer Science
"... A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all cons ..."
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Cited by 34 (4 self)
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A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutiveones orderings of the columns of a matrix that has the consecutiveones property. We give an analogous structure, called a PC tree, for representing all circularones orderings of the columns of a matrix that has the circularones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1
Finding Four Independent Trees
"... Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(V  2) algorithm for finding three independent spanning trees in a 3 ..."
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Cited by 4 (0 self)
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Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(V  2) algorithm for finding three independent spanning trees in a 3connected graph. In this paper we present an O(V  3) algorithm for finding four independent spanning trees in a 4connected graph. We make use of chain decompositions of 4connected graphs. ∗ Partially supported by NSF VIGRE grant † Supported by CNPq (Proc: 200611/003) – Brazil ‡ Partially supported by NSF grant DMS0245530 and NSA grant MDA9040310052
Apptopinv  user’s guide
, 2003
"... The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NPcomplete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on HopcroftTarjan planarity testing algorit ..."
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Cited by 1 (1 self)
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The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NPcomplete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on HopcroftTarjan planarity testing algorithm, the spanningtree heuristic and various algorithms based on the cactustree heuristic. Apptopinv contains also a simulated annealing algorithm that can be used to improve the solutions obtained from other heuristics. Most of the heuristics have also a greedy version. We have implemented graph generators for complete graphs, complete kpartite graphs, complete hypercubes, random graphs, random maximum planar and outerplanar graphs and random regular graphs. Apptopinv supports three different graph file formats. Apptopinv is written in C++ programming language for Linuxplatform and GCC 2.95.3 compiler. To compile the program, a commercial LEDA algorithm
Intervalbased graph representations for efficient web querying. Diplomarbeit/diploma thesis, LudwigMaxmilians University Munich (2009) 2 Labeling RDF Graphs for Linear Time and Space Querying
"... Hiermit versichere ich, dass ich diese Diplomarbeit selbständig verfasst habe. Ich habe dazu keine anderen als die angegebenen Quellen und Hilfsmittel verwendet. ..."
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Hiermit versichere ich, dass ich diese Diplomarbeit selbständig verfasst habe. Ich habe dazu keine anderen als die angegebenen Quellen und Hilfsmittel verwendet.
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
A planarity test via construction sequences
 CoRR
"... Abstract. Lineartime algorithms for testing the planarity of a graph are well known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler lineartime tests. We give a conceptually simple reduction from planarity testing to the problem of c ..."
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Abstract. Lineartime algorithms for testing the planarity of a graph are well known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler lineartime tests. We give a conceptually simple reduction from planarity testing to the problem of computing a certain construction of a 3connected graph. This implies a lineartime planarity test. Our approach is radically different from all previous lineartime planarity tests; as key concept, we maintain a planar embedding that is 3connected at each point in time. The algorithm computes a planar embedding if the input graph is planar and a Kuratowskisubdivision otherwise. 1
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).
Correcting and Implementing the PCtree Planarity Algorithm
"... 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. Recent research eorts have produced new algorithms for solving planarityrelated problems. Shih and Hsu proposed a lineartime algorithm based on a data st ..."
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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. Recent research eorts have produced new algorithms for solving planarityrelated problems. Shih and Hsu proposed a lineartime algorithm based on a data structure they named PCtree, which is similar to but much simpler than a PQtree. However, their presentation does not explain in detail how to implement the routines that manipulate a PCtree, and there are some nontrivial correctness and runtime issues that were not addressed in their paper. So it is far from trivial to derive a proper lineartime implementation from their description. This paper presents additions to the theoretical framework of the PCtree algorithm that are necessary to achieve correctness and linear running time. A lineartime implementation that addresses the issues raised in this paper was developed in the LEDA platform and is available.
Nonseparating planar chains in 4connected graphs
, 2002
"... In this paper, we describe an O(V (G)  2) algorithm for finding a “nonseparating planar chain” in a 4connected graph G, which will be used to decompose an arbitrary 4connected graph into “planar chains”. This work was motivated by the study of a multitree approach to reliability in distributed ..."
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In this paper, we describe an O(V (G)  2) algorithm for finding a “nonseparating planar chain” in a 4connected graph G, which will be used to decompose an arbitrary 4connected graph into “planar chains”. This work was motivated by the study of a multitree approach to reliability in distributed networks, as well as the study of nonseparating induced paths in highly connected graphs.