A 0-1 matrix has the consecutive-ones property if its columns can be ordered so that the ones in every row are consecutive. It has the circular-ones 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 consecutive-ones orderings of the columns of a matrix that has the consecutive-ones property. We give an analogous structure, called a PC tree, for representing all circular-ones orderings of the columns of a matrix that has the circular-ones 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

The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NP-complete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on Hopcroft-Tarjan planarity testing algorithm, the spanning-tree heuristic and various algorithms based on the cactus-tree 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 Linux-platform and GCC 2.95.3 compiler. To compile the program, a commercial LEDA algorithm

Hiermit versichere ich, dass ich diese Diplomarbeit selbständig verfasst habe. Ich habe dazu keine anderen als die angegebenen Quellen und Hilfsmittel verwendet.

We give a linear-time 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 non-planar graph. Our algorithm keeps track of possible circular edge orderings in a partial embedding by using a reinterpretation of Booth and Lueker’s PQ-tree data structure. This is a classic data structure that represents certain sets of permutation and gives linear-time algorithms for various matrix and graph ordering problems. We show that our reinterpretation of PQ-trees gives exactly the PC-trees of Shih and Hsu. We give a simpler and more symmetric implementation of PQ-tree reduction. This simplifies various applications and leads to an efficient algorithm for a generalization of the consecutive and circular ones problems. 1

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).

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 planarity-related problems. Shih and Hsu proposed a linear-time algorithm based on a data structure they named PC-tree, which is similar to but much simpler than a PQ-tree. However, their presentation does not explain in detail how to implement the routines that manipulate a PC-tree, and there are some nontrivial correctness and run-time issues that were not addressed in their paper. So it is far from trivial to derive a proper linear-time implementation from their description. This paper presents additions to the theoretical framework of the PC-tree algorithm that are necessary to achieve correctness and linear running time. A linear-time implementation that addresses the issues raised in this paper was developed in the LEDA platform and is available.

In this paper, we describe an O(|V (G) | 2) algorithm for finding a “non-separating planar chain” in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into “planar chains”. This work was motivated by the study of a multi-tree approach to reliability in distributed networks, as well as the study of non-separating induced paths in highly connected graphs.

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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 fixed-parameter tractable algorithm for minimizing the number of crossings over all d-ary 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