Linkage realization in polygonal domains is W[1]-hard parameterized by graph size and NP-hard for paths with prescribed endpoints, with a linear-time algorithm for short paths in convex polygons.
Journal of Graph Algorithms and Applications11(1), 259–276 (2007)
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Realizing Planar Linkages in Polygonal Domains
Linkage realization in polygonal domains is W[1]-hard parameterized by graph size and NP-hard for paths with prescribed endpoints, with a linear-time algorithm for short paths in convex polygons.