Untangling Circular Drawings: Algorithms and Complexity

TL;DR

This paper investigates the problem of transforming non-planar circular drawings of outerplanar graphs into planar ones by vertex shifting, establishing NP-completeness and providing bounds and algorithms for special cases.

Contribution

It proves NP-completeness of the Circular Untangling problem and offers tight bounds and polynomial algorithms for almost-planar circular drawings.

Findings

NP-completeness of Circular Untangling.

Tight upper bound of shift$( ext{delta}_G)$ for outerplanar graphs.

Polynomial-time algorithm for almost-planar drawings.

Abstract

We consider the problem of untangling a given (non-planar) straight-line circular drawing $\delta_G$ of an outerplanar graph $G=(V, E)$ into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph $G$, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift$(\delta_G)$ as the minimum number of vertices that are required to be shifted in order to resolve all crossings of $\delta_G$. We show that the problem Circular Untangling, asking whether shift$(\delta_G) \le K$ for a given integer $K$, is NP-complete. For $n$-vertex outerplanar graphs, we obtain a tight upper bound of shift$(\delta_G) \le n - \lfloor\sqrt{n-2}\rfloor -2$. Based on these results we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case, we provide a tight upper bound shift$(\delta_G) \le \lfloor \frac{n}{2} \rfloor-1$ and present a constructive polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.