Obfuscated Drawings of Planar Graphs

Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding of the vertex set of G into the plane. Let fix(G, ) be the maximum integer k such that there exists a crossing-free redrawing of G which keeps k vertices fixed, i.e., there exist k vertices v1, . . . , vk of G such that (vi) = (vi) for i = 1, . . . , k. We give examples of planar graphs G along with a drawing for which fix(G, ) = O( n). In fact, such a drawing exists even if it is presupposed that the vertices occupy any prescribed set of points on the boundary of a convex body. We also consider the parameter obf (G) of a graph G which is equal to the maximum number of edge crossings over all straight line drawings of G. We give examples of planar graphs with obf (G) ( 9 4 - o(1))n2 and prove that obf (T) ( 13 8 - o(1))n2 for every triangulation T. We also show that a given triangul...

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Mihyun Kang, Oleg Pikhurko, Alexander Ravsky, Mathias Schacht, Oleg Verbitsky
• Keywords: CORR 2008, Education, Injective Embedding, Planar Graphs, Straight Line Segments