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In this article, we discussed the use of various spatial tessellations to determine, in the framework of partitioning policies, optimal workload share in a mobile robotic network. We also proposed efficient and spatially distributed algorithms for achieving some of these tessellations with minimum or no communication between the agents. Because of space limitations, we have not reported results of numerical experiments in this article but provided bibliographic references to publications containing such results and further details. It is interesting to note that these tessellations appear while considering different variations of the same basic problem (DTRP). It is then natural to investigate the existence of a single objective function, whose optima correspond to the various tessellations under these different variations. The game theory approach seems to be a promising one.