Untangling circular drawings: Algorithms and complexity

We consider the problem of untangling a given (non-planar) straight-line circular drawing δG an outerplanar graph G=(V,E) into planar G by shifting minimum number vertices to new position on circle. For G, it is obvious that such crossing-free always exists and we define shift∘(δG) as are required be shifted in order resolve all crossings δG. show Circular Untangling, asking whether shift∘(δG)≤K for integer K, NP-complete. n-vertex graphs, obtain tight upper bound shift∘(δG)≤n−⌊n−2⌋−2. Moreover, study Untangling almost-planar drawings, which single edge involved crossings. this problem, provide shift∘(δG)≤⌊n2⌋−1 present O(n2)-time algorithm compute drawings.