Untangling circular drawings: Algorithms and complexity

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 of G by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is obvious that such a crossing-free circular drawing always exists and we define the circular shifting number ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)$ 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 ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)\le K$ for a given integer K, is NP-complete. For n-vertex outerplanar graphs, we obtain a tight upper bound of ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)\le n-\lfloor \sqrt{n-2}\rfloor -2$. Moreover, we study the Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all of the crossings. For this problem, we provide a tight upper bound ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)\le \lfloor \frac{n}{2}\rfloor -1$ and present an $O\left(n^2\right)$-time algorithm to compute the circular shifting number of almost-planar drawings.