Untangling planar graphs from a specified vertex position - Hard cases
  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 $\pi$ of the vertex set of $G$ into the plane. We prove that a wheel graph $W_n$ admits a drawing $\pi$ such that, if one wants to eliminate edge crossings by shifting vertices to new positions in the plane, then at most $(2+o(1))\sqrt n$ of all $n$ vertices can stay fixed. Moreover, such a drawing $\pi$ exists even if it is presupposed that the vertices occupy any prescribed set of points in the plane. Similar questions are discussed for other families of planar graphs. 
Mihyun Kang, Oleg Pikhurko, Alexander Ravsky, Mathias Schacht, Oleg   Verbitsky