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 Pn with itself has been used earlier to develop an approximation algorithm for (n2 − 1)-puzzle
@article{bwmeta1.element.doi-10_7151_dmgt_1726,
author = {Biswajit Deb and Kalpesh Kapoor},
title = {Motion planning in cartesian product graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {207-221},
zbl = {1290.05125},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1726}
}
Biswajit Deb; Kalpesh Kapoor. Motion planning in cartesian product graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 207-221. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1726/
[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[Crossref] | Zbl 1145.68471
[2] B. Deb, K. Kapoor and S. Pati, On mrj reachability in trees Discrete Math., Alg. and Appl. 4 (2012). doi:10.1142/S1793830912500553[Crossref]
[3] T. Feder, Stable Networks and Product Graphs (Memoirs of the American Mathe- matical Society, 1995). doi:10.1090/memo/0555[Crossref]
[4] R. Hammack, W. Imrich and S. Klaˇzar, Handbook of Product Graphs, Second Edition (CRC Press, New York, 2011). | Zbl 1283.05001
[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[Crossref]
[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[Crossref]
[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[Crossref]
[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[Crossref]
[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[Crossref]
[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[Crossref] | Zbl 0704.68057
[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 [Crossref]