Untangling planar graphs from a specified vertex position - Hard cases

TL;DR

This paper investigates the limitations of fixing vertex positions in straight-line drawings of planar graphs, showing that for wheel graphs, only about ew vertices can remain fixed without crossings, even with prescribed positions.

Contribution

It establishes bounds on the number of vertices that can stay fixed in crossing-free drawings of planar graphs, including wheel graphs, under various vertex placement constraints.

Findings

At most (2+o(1))loorloor n vertices can stay fixed in crossing-free drawings of wheel graphs.

Such fixed-vertex drawings exist even when vertices are placed at arbitrary prescribed points.

The results extend to other families of planar graphs.

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.