Untangling circular drawings: Algorithms and complexity

We consider the problem of untangling a given (non-planar) straight-line circular drawing δ_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 shift^∘(δ_G) as the minimum number of vertices that are required to be shifted in order to resolve all crossings of δ_G. We show that the problem CIRCULAR UNTANGLING, asking whether shift^∘(δ_G)≤K for a given integer K, is NP-complete. For n-vertex outerplanar graphs, we obtain a tight upper bound of shift^∘(δ_G)≤n−⌊n−2⌋−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 [Formula presented] and present an O(n²)-time algorithm to compute the circular shifting number of almost-planar drawings.