A crossing-free morph is a continuous deformation between two graph drawings that preserves straight-line pairwise noncrossing edges. Motivated by applications in 3D morphing problems, we initiate the study of morphing graph drawings in the plane in the presence of stationary point obstacles, which need to be avoided throughout the deformation. As our main result, we prove that it is NP-hard to decide whether such an obstacle-avoiding 2D morph between two given drawings of the same graph exists. In fact, this statement remains true even in the severely restricted special case where only three vertices have to change positions. This is in sharp contrast to the classical case without obstacles, where there is an efficiently verifiable (necessary and sufficient) criterion for the existence of a morph. Further, we provide several combinatorial results related to conditions under which the existence of a morph between two drawings of a graph can or cannot be prevented by the placement of a given number of point obstacles.

翻译：无交叉变形是指两个图形绘制之间的连续形变过程，其保持直线边对之间的非交叉特性。受三维变形问题应用的启发，我们首次研究了在平面内存在静止点障碍物情况下的图形绘制变形问题，这些障碍物需要在整个形变过程中被规避。作为主要研究成果，我们证明了判定两个给定同图绘制间是否存在此类避障二维变形是NP难问题。事实上，该结论即使在仅需改变三个顶点位置的严格受限特殊情况下依然成立。这与经典无障碍物情形形成鲜明对比——后者存在可高效验证的变形存在性充要条件。此外，我们提出了若干组合性质结论，涉及给定数量点障碍物的布置能否阻止或确保两个图形绘制间变形存在的条件。