@phdthesis{Kryven2022,
  author    = {Kryven, Myroslav},
  title     = {Optimizing Crossings in Circular-Arc Drawings and Circular Layouts},
  isbn      = {978-3-95826-174-7},
  doi       = {10.25972/WUP-978-3-95826-175-4},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-245960},
  school      = {Universit{\"a}t W{\"u}rzburg},
  pages     = {viii, 129},
  year      = {2022},
  abstract  = {A graph is an abstract network that represents a set of objects, called vertices, and relations between these objects, called edges. Graphs can model various networks. For example, a social network where the vertices correspond to users of the network and the edges represent relations between the users. To better see the structure of a graph it is helpful to visualize it. The research field of visualizing graphs is called Graph Drawing. A standard visualization is a node-link diagram in the Euclidean plane. In such a representation the vertices are drawn as points in the plane and edges are drawn as Jordan curves between every two vertices connected by an edge. Edge crossings decrease the readability of a drawing, therefore, Crossing Optimization is a fundamental problem in Graph Drawing. Graphs that can be drawn with few crossings are called beyond-planar graphs. The topic that deals with definition and analysis of beyond-planar graphs is called Beyond Planarity and it is an important and relatively new research area in Graph Drawing. In general, beyond planar graphs posses drawings where edge crossings are restricted in some way. For example, the number of crossings may be bounded by a constant independent of the size of the graph. Crossings can also be restricted locally by, for example, restricting the number of crossings per edge, restricting the number of pairwise crossing edges, or bounding the crossing angle of two edges in the drawing from below. This PhD thesis defines and analyses beyond-planar graph classes that arise from such local restrictions on edge crossings.},
  subject      = {Graphenzeichnen},
  language  = {en}
}
@phdthesis{Fink2014,
  author    = {Fink, Martin},
  title     = {Crossings, Curves, and Constraints in Graph Drawing},
  publisher = {W{\"u}rzburg University Press},
  isbn      = {978-3-95826-002-3 (print)},
  doi       = {10.25972/WUP-978-3-95826-003-0},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-98235},
  school      = {W{\"u}rzburg University Press},
  pages     = {222},
  year      = {2014},
  abstract  = {In many cases, problems, data, or information can be modeled as graphs. Graphs can be used as a tool for modeling in any case where connections between distinguishable objects occur. Any graph consists of a set of objects, called vertices, and a set of connections, called edges, such that any edge connects a pair of vertices. For example, a social network can be modeled by a graph by transforming the users of the network into vertices and friendship relations between users into edges. Also physical networks like computer networks or transportation networks, for example, the metro network of a city, can be seen as graphs. For making graphs and, thereby, the data that is modeled, well-understandable for users, we need a visualization. Graph drawing deals with algorithms for visualizing graphs. In this thesis, especially the use of crossings and curves is investigated for graph drawing problems under additional constraints. The constraints that occur in the problems investigated in this thesis especially restrict the positions of (a part of) the vertices; this is done either as a hard constraint or as an optimization criterion.},
  subject      = {Graphenzeichnen},
  language  = {en}
}
@article{BuchinBuchinByrkaetal.2012,
  author    = {Buchin, Kevin and Buchin, Maike and Byrka, Jaroslaw and N{\"o}llenburg, Martin and Okamoto, Yoshio and Silveira, Rodrigo I. and Wolff, Alexander},
  title     = {Drawing (Complete) Binary Tanglegrams},
  series = {Algorithmica},
  volume    = {62},
  journal   = {Algorithmica},
  doi       = {10.1007/s00453-010-9456-3},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-124622},
  pages     = {309-332},
  year      = {2012},
  abstract  = {A binary tanglegram is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an O(n 3)-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.},
  language  = {en}
}