Let G be an undirected graph with n vertices. Assume that a robot is placed on a vertex and n − 2 obstacles are placed on the other vertices. A vertex on which neither a robot nor an obstacle is placed is said to have a hole. Consider a single player game in which a robot or obstacle can be moved to adjacent vertex if it has a hole. The objective is to take the robot to a fixed destination vertex using minimum number of moves. In general, it is not necessary that the robot will take a shortest path between the source and destination vertices in graph G. In this article we show that the path traced by the robot coincides with a shortest path in case of Cartesian product graphs. We give the minimum number of moves required for the motion planning problem in Cartesian product of two graphs having girth 6 or more. A result that we prove in the context of Cartesian product of P_{n} with itself has been used earlier to develop an approximation algorithm for (n^{2} − 1)-puzzle
[1] G. Calinescu A. Dumitrescu and J. Pach Reconfigurations in graphs and grids in: Latin American Theoretical Informatics Conference Lecture Notes in Computer Science 3887 (2006) 262-273. doi:10.1007/11682462 27
[2] B. Deb K. Kapoor and S. Pati On mrj reachability in trees Discrete Math. Alg. and Appl. 4 (2012). doi:10.1142/S1793830912500553
[3] T. Feder Stable Networks and Product Graphs (Memoirs of the American Mathe- matical Society 1995). doi:10.1090/memo/0555
[4] R. Hammack W. Imrich and S. Klaˇzar Handbook of Product Graphs Second Edition (CRC Press New York 2011).
[5] F. Harary Graph Theory (Addison-Wesley Reading MA 1969).
[6] W. Imrich and S. Klavˇzar Product Graphs Structure and Recognition (John Wiley & Sons Inc. New York 2000).
[7] W. Woolsey Johnson and W.E. Story Notes on the ”15” puzzle Amer. J. Math. 2 (1879) 397-404. doi:10.2307/2369492
[8] D. Kornhauser G. Miller and P. Spirakis Coordinating pebble motion on graphs the diameter of permutation groups and applications in: Annual Symposium on Foundations of Computer Science IEEE Computer Society (1984) 241-250. doi:10.1109/SFCS.1984.715921
[9] E. Masehian and A.H. Nejad Solvability of multi robot motion planning problems on trees in: IEEE/RSJ International Conference on Intelligent Robots and Systems Piscataway NJ USA (2009) IEEE Press. 5936-5941. doi:10.1109/IROS.2009.5354148
[10] C.H. Papadimitriou P. Raghavan M. Sudan and H. Tamaki Motion planning on a graph in: FOCS’94 (1994) 511-520. doi:10.1109/SFCS.1994.365740
[11] I. Parberry A real-time algorithm for the (n2 −1)-puzzle Inform. Process. Lett. 56 (1995) 23-28. doi:10.1016/0020-0190(95)00134-X
[12] D. Ratner and M.K. Warmuth The (n2 −1)-puzzle and related relocation problems J. Symbolic Comput. 10 (1990) 111-137. doi:10.1016/S0747-7171(08)80001-6
[13] D. Ratner and M.K. Warmuth Finding a shortest solution for the n × n extension of the 15-puzzle is intractable in: AAAI(1986) 168-172.
[14] R.M. Wilson Graph puzzles homotopy and the alternating group J. Combin. The- ory (B) 16 (1974) 86-96. doi:10.1016/0095-8956(74)90098-7