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Given a planar graph $G$, we consider drawings of $G$ in the plane whereedges are represented by straight line segments (which possibly intersect).Such a drawing is specified by an injective embedding $\pi$ of the vertex setof $G$ 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 newpositions in the plane, then at most $(2+o(1))\sqrt n$ of all $n$ vertices canstay fixed. Moreover, such a drawing $\pi$ exists even if it is presupposedthat the vertices occupy any prescribed set of points in the plane. Similarquestions are discussed for other families of planar graphs.

Untangling planar graphs from a specified vertex position - Hard cases