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Another characterisation of planar graphs
"... A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph. 1 ..."
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A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph. 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.
An algorithm for embedding graphs in the torus
"... An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applyi ..."
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An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applying a recent result of Fiedler, Huneke, Richter, and Robertson [5] about the genus of graphs in the projective plane, and simplifies other steps on the expense of losing linear time complexity. 1
A Faster Algorithm for Torus Embedding
, 2004
"... Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and di ..."
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Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the wellstudied problem of finding the complete set of torus obstructions.
Recognizing Compound Planarity of Graphs
"... this paper, we introduce a practical and simple graph model called clustered graphs. We study, in particular, the planarity problem associated with this graph model. A clustered graph consists of a graph G and a recursive partitioning of the vertices of G. Each partition is known as a cluster of a ..."
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this paper, we introduce a practical and simple graph model called clustered graphs. We study, in particular, the planarity problem associated with this graph model. A clustered graph consists of a graph G and a recursive partitioning of the vertices of G. Each partition is known as a cluster of a subset of the vertices of G. Clustering appears in the diagrams produced in a wide number of applications areas, such as software engineering [22], knowledge representation [14], software visualization [21], idea organization [15], VLSI design [10], and general divide and conquer problem solving methodologies. Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be NPhard [8]. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist [12, 17, 5, 6]. In this paper, we introduce compound planarity (cplanarity), the planarity of clustered graphs. In a drawing of a clustered graph, vertices and edges are drawn as points and curves as usual. Clusters are drawn as simple closed curves that define closed regions of the plane. The region for each cluster contains the drawing of the subgraph induced by its vertices and no other vertices. A region for a cluster contains the regions for all its subclusters and does not intersect the region for any other cluster. A clustered graph is compoundplanar (cplanar) if it has a drawing with no crossings between distinct edges, or crossings between an edge and a region. Note that the planarity of the underlying graph does not
Purity, Impurity and Efficiency in Graph Algorithms
"... Introduction This chapter initially considers pure lazy functional languages: their philosophy, advantages and disadvantages. We then examine how to develop efficient lazy functional programs. One way to achieve efficiency is to introduce impurities. In the final section the two schools of lazy fun ..."
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Introduction This chapter initially considers pure lazy functional languages: their philosophy, advantages and disadvantages. We then examine how to develop efficient lazy functional programs. One way to achieve efficiency is to introduce impurities. In the final section the two schools of lazy functional programming, pure and impure, are assessed. The assessment centres around two partial implementations of the Hopcroft Tarjan graph planarity algorithm. Profiling tools are used to make an experimental comparison and optimisation of each program. 4.1 Lazy Functional Programming In his book [42] Reade suggests that the user of a traditional imperative language is required to do the following: 1. describe the result to be computed; 2. impose an order on the steps required in the computation; 3. create and destroy, as required, any data structures used by the computation. 74 The first item is concerned with the extensional prope
Cluster Planarity Testing for the Case of Not Necessarily Connected Clusters
, 2006
"... The central topic of this thesis are criteria and tests which reveal whether a given clustered graph allows an embedding in the plane for which no edges and clusters intersect. Together with their definition in 1996, a notion of planarity was presented for clustered graphs, as well as an algorithm w ..."
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The central topic of this thesis are criteria and tests which reveal whether a given clustered graph allows an embedding in the plane for which no edges and clusters intersect. Together with their definition in 1996, a notion of planarity was presented for clustered graphs, as well as an algorithm which tests this planarity for a given clustered graph in linear time. The algorithm however expects each cluster to be connected. For general clustered graphs, no efficient algorithm is yet known, neither is the computational complexity of the problem. This work presents algorithms which extend the class of clustered graphs for which planarity can be tested in polynomial time. A second part considers a weak form of planarity, and shows that a polynomial time test for this form also yields a polynomial time test for the classical definition. Furthermore, an attempt is made, by means of a characterization of the weak realizability problem in terms of forbidden subgraphs, to gain a similar characterization of the weak form of cluster planarity.