A counterexample to the Alexopoulos-Griffin path planning algorithm

The planar stationary-obstacle path-planning problem for polygonal obstacles has been correctly and completely solved by T. Lozano-Perez and M. Wesley (1979), i.e., a global, optimal algorithm was provided which requires O(μ²logμ) computation time, where μ is the number of obstacle-faces in the scene. That algorithm is known as the VGRAPH algorithm. Two variants of VGRAPH have been developed to solve the same problem in O(μ²) computation time. Our paper discusses a recent algorithm proposed by C. Alexopoulos and P.M. Griffin (1992), called V*GRAPH, which also claims to provide an optimal solution. We demonstrate by counter-example that V*GRAPH is neither global nor optimal