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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
Contractions, Removals and How to Certify 3Connectivity in Linear Time
"... One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the exis ..."
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One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K4, such that every intermediate graph is 3vertexconnected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(V  2) to O(E). This result has a number of consequences; an important one is a new lineartime test of 3connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in the last years. The test is conceptually different from wellknown lineartime 3connectivity tests and uses a certificate that is easy to verify in time O(E). We show how to extend the results to an optimal certifying test of 3edgeconnectivity. 1
Algorithm and Experiments in Testing Planar . . .
, 2004
"... We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determi ..."
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We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determine the conditions in which the implemented algorithm outperforms other graph matchers, which do not impose topological restrictions on graphs. We report experiments with our planar graph matcher tested against McKay’s, Ullmann’s, and SUBDUE’s (a graphbased data mining system) graph matchers.
Cluster Planarity Testing for the Case of Not Necessarily Connected Clusters
"... 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.
Novel Method for Improving the Exact Matching of the Molecular Graphs
"... Abstract — one way of determining chemical and biological reactivity of a newly found compound is by searching database for structurally similar molecules. Graph theory concepts are being used for molecular matching. Molecular matching is of two kinds, like, complete matching and partial matching (l ..."
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Abstract — one way of determining chemical and biological reactivity of a newly found compound is by searching database for structurally similar molecules. Graph theory concepts are being used for molecular matching. Molecular matching is of two kinds, like, complete matching and partial matching (like searching for functional groups). In this paper we propose an efficient way of pruning the large molecular databases in various stages in order to do coarse filtering which uses bitstring manipulation, histogram filtering and dimensionality reduction to prune some or most of the database molecules. Then exact matching with the query molecule is performed through fine filtering on the remaining molecules in order to find the exact match for the query molecule from the pruned database using more expensive graph isomorphism algorithm. In this way search time can be reduced significantly. I.
An Algorithm for 3Dbiplanar Graph Drawing
"... We introduce the concept of 3Dbiplanar drawing in which we partition a graph into two planar induced subgraphs. Our goal is to find such a partition with the minimum number of edges between the two partitions. We prove that this problem is NPcomplete and present a randomized parameterized algorith ..."
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We introduce the concept of 3Dbiplanar drawing in which we partition a graph into two planar induced subgraphs. Our goal is to find such a partition with the minimum number of edges between the two partitions. We prove that this problem is NPcomplete and present a randomized parameterized algorithm with O(n k) time, where k is the ratio of the optimal solution to the mincut size of the graph. 1
SUMMARY
"... A wide number of practical applications would benefit from automatically generated graphical representations of database schemas, in which tables are represented by boxes, and table attributes correspond to distinct stripes inside each table. Links, connecting attributes of two different tables, rep ..."
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A wide number of practical applications would benefit from automatically generated graphical representations of database schemas, in which tables are represented by boxes, and table attributes correspond to distinct stripes inside each table. Links, connecting attributes of two different tables, represent referential constraints or join relationships, and may attach arbitrarily to the left or to the righthand side of the stripes representing the attributes. To our knowledge no drawing technique is available to automatically produce diagrams in such a strongly constrained drawing convention. In this paper we provide a polynomial time algorithm for solving this problem, and test its efficiency and effectiveness against a large test suite. Also, we describe an implementation of a system that uses such an algorithm and we study the main methodological problems we faced in developing such a technology. Copyright © 2002 John Wiley &Sons,Ltd. KEY WORDS: graph drawing; algorithm engineering; drawing standard; orthogonal drawing; database schema visualization; upward drawing