The routing number of an s-graph arises in the social-law approach to the problem of coordination ofrobots moving on a network.Onn and Sperber showed that the determination of the routing number of an s-graph is NP-complete and further posed the question of whether planar graphs always admit routing numbers that can be bounded in terms of their radius.In this paper

we introduce the routing number of the directed s-graph and show that it is equal to the perimeter (i.e. the length of a longest directed cycle) of the directed s-graph.This implies that the routing number of an undirected s-graph is equal to the minimum perimeter among all its orientations.Using this result

we give a counter example to a question of Onn et al.and further show that the routing number of the product graph

a graph frequently used as a network

is equal to its radius and hence can be determined in polynomial time.