Geodesic
Motion planning in Cartesian product graphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 207-221.

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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 P_n with itself has been used earlier to develop an approximation algorithm for (n^2 − 1)-puzzle
Keywords: robot motion in a graph, Cartesian product of graphs 
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Deb, Biswajit; Kapoor, Kalpesh. Motion planning in Cartesian product graphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 207-221. http://geodesic.mathdoc.fr/item/DMGT_2014_34_2_a0/

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