Minimally distant sets of lattice points (Q1260773)

In this paper, authors consider the problem of finding two sets of given cardinalities \(x\) and \(y\) in certain grid graphs, to minimize the maximum distance between two nodes respectively in the two sets. This minimax distance is denoted by \(D(x,y)\). For grids that are a product of chains of even edge length, and for the \(n\)-cube, the problem is solved completely. The answer in these cases is that both sets are initial segments of some ordering of vertices. An open question proposed in the end of this paper is as follows: For which chain products do orderings of the vertices exist such that initial segments minimize \(D(x,x)\), for all \(x\)?