dc.contributor.author Chan, Hubert en dc.date.accessioned 2006-08-22 14:25:39 (GMT) dc.date.available 2006-08-22 14:25:39 (GMT) dc.date.issued 2003 en dc.date.submitted 2003 en dc.identifier.uri http://hdl.handle.net/10012/1090 dc.description.abstract We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NP-complete. Parameterized complexity is a technique used to find efficient algorithms for hard problems, and in particular, NP-complete problems. The main idea is that the complexity of an algorithm can be constrained, for the most part, to a parameter that describes some aspect of the problem. If the parameter is fixed, the algorithm will run in polynomial time. In this thesis, we investigate contracting an edge in an upward planar graph that has a specified embedding, and show that we can determine whether or not the resulting embedding is upward planar given the orientation of the clockwise and counterclockwise neighbours of the given edge. Using this result, we then show that under certain conditions, we can join two upward planar graphs at a vertex and obtain a new upward planar graph. These two results expand on work done by Hutton and Lubiw. Finally, we show that a biconnected graph has at most k!8k-1 planar embeddings, where k is the number of triconnected components. By using an algorithm by Bertolazzi et al. that tests whether a given embedding is upward planar, we obtain a parameterized algorithm, where the parameter is the number of triconnected components, for testing the upward planarity of a biconnected graph. This algorithm runs in O(k!8kn3) time. en dc.format application/pdf en dc.format.extent 575170 bytes dc.format.mimetype application/pdf dc.language.iso en en dc.publisher University of Waterloo en dc.rights Copyright: 2003, Chan, Hubert. All rights reserved. en dc.subject Computer Science en dc.subject algorithms en dc.subject graph theory en dc.subject graph drawing en dc.subject parameterized complexity en dc.subject upward planarity en dc.title A Parameterized Algorithm for Upward Planarity Testing of Biconnected Graphs en dc.type Master Thesis en dc.pending false en uws-etd.degree.department School of Computer Science en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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