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arXiv · 0803.0858

Untangling planar graphs from a specified vertex position - Hard cases

Mihyun Kang·Oleg Pikhurko·Alexander Ravsky·Mathias Schacht·Oleg Verbitsky
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Abstract

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.

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Mihyun Kang, Oleg Pikhurko, Alexander Ravsky, Mathias Schacht, Oleg Verbitsky. 2008-03-06. Untangling planar graphs from a specified vertex position - Hard cases. https://doi.org/10.1016/j.dam.2011.01.011

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