Untangling Polygons and Graphs
- 69 Downloads
Untangling is a process in which some vertices in a drawing of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C n while keeping Ω(n 2/3) vertices fixed.
For any connected graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree, and diameter of G. One consequence is that every 3-connected planar graph has a drawing δ such that at most O((nlog n)2/3) vertices are fixed in every untangling of δ.
KeywordsGraph drawing Planarity Straight-line drawing Untangling Polygons
Unable to display preview. Download preview PDF.
- 3.Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935) Google Scholar
- 4.Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged, Acta Sci. Math. 11, 229–233 (1948) Google Scholar
- 6.Kang, M., Pikhurko, O., Ravsky, A., Schacht, M., Verbitsky, O.: Obfuscated drawings of planar graphs. 0803.0858
- 8.Ravsky, A., Verbitsky, O.: On collinear sets in straight line drawings. 0806.0253
- 11.Wagner, K.: Bemerkungen zum Vierfarbenproblem. Jahresber. Deutsch. Math. Verein. 46, 26–32 (1936) Google Scholar