On Geometric Drawings of Graphs
| dc.contributor.author | Arroyo Guevara, Alan Marcelo | |
| dc.date.accessioned | 2018-04-18T16:26:51Z | |
| dc.date.available | 2018-04-18T16:26:51Z | |
| dc.date.issued | 2018-04-18 | |
| dc.date.submitted | 2018-04-17 | |
| dc.description.abstract | This thesis is about geometric drawings of graphs and their topological generalizations. First, we study pseudolinear drawings of graphs in the plane. A pseudolinear drawing is one in which every edge can be extended into an infinite simple arc in the plane, homeomorphic to $\mathbb{R}$, and such that every two extending arcs cross exactly once. This is a natural generalization of the well-studied class of rectilinear drawings, where edges are straight-line segments. Although, the problem of deciding whether a drawing is homeomorphic to a rectilinear drawing is NP-hard, in this work we characterize the minimal forbidden subdrawings for pseudolinear drawings and we also provide a polynomial-time algorithm for recognizing this family of drawings. Second, we consider the problem of transforming a topological drawing into a similar rectilinear drawing preserving the set of crossing pairs of edges. We show that, under some circumstances, pseudolinearity is a necessary and sufficient condition for the existence of such transformation. For this, we prove a generalization of Tutte's Spring Theorem for drawings with crossings placed in a particular way. Lastly, we study drawings of $K_n$ in the sphere whose edges can be extended to an arrangement of pseudocircles. An arrangement of pseudocircles is a set of simple closed curves in the sphere such that every two intersect at most twice. We show that (i) there is drawing of $K_{10}$ that cannot be extended into an arrangement of pseudocircles; and (ii) there is a drawing of $K_9$ that can be extended to an arrangement of pseudocircles, but no extension satisfies that every two pseudocircles intersects exactly twice. We also introduce the notion pseudospherical drawings of $K_n$, a generalization of spherical drawings in which each edge is a minor arc of a great circle. We show that these drawings are characterized by a simple local property. We also show that every pseudospherical drawing has an extension into an arrangement of pseudocircles where the ``at most twice'' condition is replaced by ``exactly twice''. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/13103 | |
| dc.language.iso | en | en |
| dc.pending | false | |
| dc.publisher | University of Waterloo | en |
| dc.subject | combinatorics | en |
| dc.subject | crossing numbers | en |
| dc.subject | graph theory | en |
| dc.subject | graph | en |
| dc.subject | drawing | en |
| dc.subject | complete graph | en |
| dc.subject | rectilinear | en |
| dc.subject | graph embedding | en |
| dc.subject | graph drawing | en |
| dc.title | On Geometric Drawings of Graphs | en |
| dc.type | Doctoral Thesis | en |
| uws-etd.degree | Doctor of Philosophy | en |
| uws-etd.degree.department | Combinatorics and Optimization | en |
| uws-etd.degree.discipline | Combinatorics and Optimization | en |
| uws-etd.degree.grantor | University of Waterloo | en |
| uws.contributor.advisor | Richter, Bruce | |
| uws.contributor.affiliation1 | Faculty of Mathematics | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.published.city | Waterloo | en |
| uws.published.country | Canada | en |
| uws.published.province | Ontario | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |
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