Flip Distance is in FPT time $O(n+ k \cdot c^k)$

Let T be a triangulation of a set P of n points in the plane, and let e be an edge shared by two triangles in T such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of P from T . The flip distance between two triangulations of P is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of P is at most k, for some given k ∈ N. It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time O(n + k · c), for a constant c ≤ 2 · 14, which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.