# Motion planning in cartesian product graphs

• Volume: 34, Issue: 2, page 207-221
• ISSN: 2083-5892

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## Abstract

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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

## How to cite

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Biswajit Deb, and Kalpesh Kapoor. "Motion planning in cartesian product graphs." Discussiones Mathematicae Graph Theory 34.2 (2014): 207-221. <http://eudml.org/doc/267655>.

@article{BiswajitDeb2014,
abstract = {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},
author = {Biswajit Deb, Kalpesh Kapoor},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {robot motion in a graph; Cartesian product of graphs},
language = {eng},
number = {2},
pages = {207-221},
title = {Motion planning in cartesian product graphs},
url = {http://eudml.org/doc/267655},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Biswajit Deb
AU - Kalpesh Kapoor
TI - Motion planning in cartesian product graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 2
SP - 207
EP - 221
AB - 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
LA - eng
KW - robot motion in a graph; Cartesian product of graphs
UR - http://eudml.org/doc/267655
ER -

## References

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9. [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. [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. [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]
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