Untangling planar graphs from a specified vertex position—Hard cases

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. Let $\mathrm{fix}(G,\pi )$ be the maximum integer $k$ such that there exists a crossing-free redrawing ${\pi }^{\prime }$ of $G$ which keeps $k$ vertices fixed, i.e., there exist $k$ vertices $v_1,\dots ,v_k$ of $G$ such that $\pi \left(v_i\right)={\pi }^{\prime }\left(v_i\right)$ for $i=1,\dots ,k$. Given a set of points $X$, let ${\mathrm{fix}}^X\left(G\right)$ denote the value of $\mathrm{fix}(G,\pi )$ minimized over $\pi$ locating the vertices of $G$ on $X$. The absolute minimum of $\mathrm{fix}(G,\pi )$ is denoted by $\mathrm{fix}\left(G\right)$.

For the wheel graph $W_n$, we prove that ${\mathrm{fix}}^X\left(W_n\right)\le (2+o\left(1\right))\sqrt{n}$ for every $X$. With a somewhat worse constant factor this is also true for the fan graph $F_n$. We inspect also other graphs for which it is known that $\mathrm{fix}\left(G\right)=O\left(\sqrt{n}\right)$.

We also show that the minimum value $\mathrm{fix}\left(G\right)$ of the parameter ${\mathrm{fix}}^X\left(G\right)$ is always attainable by a collinear $X$.