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Open access
Author
Date
2023Type
- Doctoral Thesis
ETH Bibliography
yes
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Abstract
This thesis revolves around the topic of beyond-planar graphs. Planar graphs are graphs that can be drawn on the plane without any crossings. Beyond-planar graphs are a generalization of planar graphs to sparse non-planar graphs. This generalization is achieved by restricting the edge crossing patterns or the number of crossings. The class of k-planar graphs, which are graphs that can be drawn in the plane with at most k crossings per edge, is a class of beyond-planar graphs.
We study fan-planar graphs in the first part of the thesis. Fan-planar graphs are graphs that admit a fan-planar drawing, that is, a drawing where all the edges crossing a specific edge e have a common endpoint and this endpoint lies on the same side of the edge e at every crossing. We study the notion of simple and non-simple topological drawings in the context of fan-planar graphs. Simple topological drawings are drawings where any pair of edges have at most one point in common including endpoints. We prove that every non-simple fan-plane drawing can be redrawn into a simple fan-plane drawing of the same graph. Essentially, this proves that all the properties of simple fan-planar graphs also hold for non-simple fan-planar graphs.
In the second part of the thesis, we study the edge density of maximal 2-planar graphs, where a maximal graph refers to a graph where no edge can be added such that the graph retains its defining properties (in this case, 2-planarity). It is known that the maximum number of edges in a 2-planar graph is 5n-10. We prove that the minimum number of edges in any maximal 2-planar graphs is at least 2n. We further show that this bound is tight by constructing a maximal 2-planar graph with 2n+c edges, where c is a constant.
In the final part of the thesis, we study the relationship between different types of crossing numbers, specifically local crossing number and simple local crossing number. We consider this problem in the context of straight-line drawings as well and general drawings. We prove that the simple local crossing number of a graph can be bounded by a function of its local crossing number. Interestingly, in contrast, we discover that the rectilinear local crossing number can be unbounded, even if the graph has a constant local crossing number. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000654918Publication status
publishedExternal links
Search print copy at ETH Library
Publisher
ETH ZurichSubject
graph drawing; graph theory; Beyond planarity; Combinatorial geometry; computational geometryOrganisational unit
03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)
Funding
171681 - Arrangements and Drawings (ArrDra) (SNF)
Related publications and datasets
Is new version of: https://doi.org/10.3929/ethz-b-000621956
Is new version of: https://doi.org/10.3929/ethz-b-000652607
Is new version of: http://hdl.handle.net/20.500.11850/465025
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ETH Bibliography
yes
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