We consider the problem of untangling a given (non-planar) straight-line
circular drawing δG​ 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(δG​) as the minimum number of vertices that are
required to be shifted in order to resolve all crossings of δG​. We show
that the problem Circular Untangling, asking whether shift(δG​)≤K
for a given integer K, is NP-complete. For n-vertex outerplanar graphs, we
obtain a tight upper bound of shift(δG​)≤n−⌊n−2​⌋−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(δG​)≤⌊2n​⌋−1 and present a constructive polynomial-time algorithm to
compute the circular shifting number of almost-planar drawings.Comment: 20 pages, 10 figures, extended version of ISAAC 2021 pape