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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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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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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
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.
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