un 2 00 9 How Much Work Does It Take To Straighten a Plane Graph Out ?

We prove that if one wants to make a plane graph drawing straightline then in the worst case one has to move almost all vertices. The second vesion of this e-print includes literally the first version. In addition, Appendix A gives an explicit bound on the number of fixed vertices and Appendix B gives an overview of related work. The final version appears as [KPRSV09]. We use the standard concepts of a plane graph and a plane embedding (or drawing) of an abstract planar graph (see, e.g., [1]). Given a plane graph G, we want to redraw it making all its edges straight line segments while keeping as many vertices on the spot as possible. Let shift(G) denote the smallest s such that we can do the job by shifting only s vertices. We define s(n) to be the maximum shift(G) over all G with n vertices. The function s(n) can have another interpretation closely related to a nice web puzzle called Planarity Game [3]. At the start of the game, a player sees a straight line drawing of a planar graph with many edge crossings. In a move s/he is allowed to shift one vertex to a new position; the incident edges are redrawn correspondingly (being all the time straight line segments). The objective is to obtain a crossing-free drawing. Thus, s(n) is equal to the number of moves that the player, playing optimally, is forced to make on an n-vertex game instance at the worst case. Institut für Informatik, Humboldt Universität zu Berlin, D-10099 Berlin Institut für Informatik, Humboldt Universität zu Berlin, D-10099 Berlin IAPMM, Lviv 79060, Ukraine. Supported by an Alexander von Humboldt return fellowship.