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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 37 (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 9 (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
Isomorphism of graph classes related to the circularones property
 Eprint
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Testing Simultaneous Planarity when the Common Graph is 2Connected
, 2011
"... Two planar graphs G1 and G2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a lin ..."
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Cited by 7 (1 self)
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Two planar graphs G1 and G2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a lineartime algorithm to test simultaneous planarity when the two graphs share a 2connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them, and the common subgraph is 2connected.
Untangling Tanglegrams: Comparing Trees by their Drawings ∗
, 2009
"... A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimi ..."
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Cited by 3 (0 self)
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A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimizations problems in tanglegram drawings. We show a linear time algorithm to decide if a tanglegram admits a planar embedding by a reduction to the planar graph drawing problem. This problem was also studied by Fernau, Kauffman and Poths (FSTTCS 2005). A similar reduction to a graph crossing problem also helps to solve an open problem they posed, showing a fixedparameter tractable algorithm for minimizing the number of crossings over all dary trees. For the case where one tree is fixed, we show an O(n log n) algorithm to determine the drawing of the second tree that minimizes the number of crossings. This improves the bound from earlier methods. We introduce a new optimization criterion using Spearman’s footrule distance and give an O(n 2) algorithm. We also show integer programming formulations to quickly obtain tanglegram drawings that minimize the two optimization measures discussed. We prove lower bounds on the maximum gap between the optimal solution and the heuristic of Dwyer and Schreiber (Austral. Symp. on Info. Vis. 2004) to minimize crossings. 1
Chain decompositions of 4connected graphs
, 2003
"... In this paper we give a decomposition of a 4connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2connected graph. We also give an O(V(G)² E(G)) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improv ..."
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Cited by 2 (2 self)
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In this paper we give a decomposition of a 4connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2connected graph. We also give an O(V(G)² E(G)) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to O(V(G)³). This decomposition will be used to find four independent spanning trees in a 4connected graph.
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
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.