dc.contributor.author Derka, Martin dc.date.accessioned 2017-08-29 14:39:13 (GMT) dc.date.available 2017-08-29 14:39:13 (GMT) dc.date.issued 2017-08-29 dc.date.submitted 2017-08-18 dc.identifier.uri http://hdl.handle.net/10012/12253 dc.description.abstract A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph en has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called \$B_2\$-VPG representations) and simultaneously two curves intersect each other at most once (those are the so-called 1-string representations). Thus, planar graphs are \$B_2\$-VPG \$1\$-string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The \$B_2\$-VPG representations of planar graphs satisfy these restrictions. We attempt to further restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate \$B_1\$-VPG representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string. dc.language.iso en en dc.publisher University of Waterloo en dc.subject string representation en dc.subject B2-VPG graph en dc.subject string graph en dc.title Restricted String Representations en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department David R. Cheriton School of Computer Science en uws-etd.degree.discipline Computer Science en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws.contributor.advisor Biedl, Therese uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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