eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-06-11
29:1
29:17
10.4230/LIPIcs.SoCG.2019.29
article
Dual Circumference and Collinear Sets
Dujmović, Vida
1
Morin, Pat
2
School of Computer Science and Electrical Engineering, University of Ottawa, Canada
School of Computer Science, Carleton University, Canada
We show that, if an n-vertex triangulation T of maximum degree Delta has a dual that contains a cycle of length l, then T has a non-crossing straight-line drawing in which some set, called a collinear set, of Omega(l/Delta^4) vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every n-vertex planar graph of maximum degree Delta has a collinear set of size Omega(n^{0.8}/Delta^4). Very recently, Dujmović et al. (SODA 2019) showed that, if S is a collinear set in a triangulation T then, for any point set X subset R^2 with |X|=|S|, T has a non-crossing straight-line drawing in which the vertices of S are drawn on the points in X. Because of this, collinear sets have numerous applications in graph drawing and related areas.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol129-socg2019/LIPIcs.SoCG.2019.29/LIPIcs.SoCG.2019.29.pdf
Planar graphs
collinear sets
untangling
column planarity
universal point subsets
partial simultaneous geometric drawings