Covering Grids by Trees

Given n points in the plane, a covering tree is a tree whose edges are line segments that jointly cover all the points. Let Gn be a n× · · · × n grid in Z. It is known that Gn can be covered by an axis-aligned polygonal path with 32n 2 + O(n) edges, thus in particular by a polygonal tree with that many edges. Here we show that every covering tree for the n points of Gn has at least (1 + c3)n 2 edges, for some constant c3 > 0. On the other hand, there exists a covering tree for the n points of Gn consisting of only n 2 +n+ 1 line segments, where each segment is either a single edge or a sequence of collinear edges. Extensions of these problems to higher dimensional grids (i.e., Gn for d ≥ 3) are also examined.