Untangling a Planar Graph

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to untangle $\delta$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate. Our hardness results extend to a version of \textsc{1BendPointSetEmbeddability}, a well-known graph-drawing problem. Further we define fix$(G,\delta)=n-shift(G,\delta)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $\delta$. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log \log n}$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $\delta_G$ of $G$ with fix$(G,\delta_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $\delta_H$ of $H$ with fix$(H,\delta_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

💡 Research Summary

The paper investigates the “untangling” problem for straight‑line drawings of planar graphs. Given a planar graph G and a straight‑line drawing δ that may contain edge crossings, the authors define shift(G, δ) as the smallest number of vertices that must be moved in order to obtain a plane (crossing‑free) drawing. Their first major contribution is a hardness proof: they show that computing shift(G, δ) exactly is NP‑hard. The reduction is from Vertex Cover; they construct a planar graph and a drawing such that any set of vertices whose movement eliminates all crossings corresponds one‑to‑one with a vertex cover of the original instance. Moreover, they prove that even approximating shift(G, δ) within any constant factor is NP‑hard, using a gap‑reduction technique. This inapproximability result extends to the well‑studied 1‑Bend Point‑Set Embeddability problem, indicating that allowing a single bend per edge does not alleviate the computational difficulty.

Having established the intrinsic hardness of minimizing vertex moves, the authors turn to the complementary measure fix(G, δ) = |V(G)| − shift(G, δ), which counts the maximum number of vertices that can remain fixed while untangling the drawing. For arbitrary planar graphs they devise a polynomial‑time algorithm that guarantees at least

\