Shortest polygonal chains covering each planar square grid

Abstract

Given any n ∈ Z + , we constructively prove the existence of covering paths and circuits in the plane which are characterized by the same link length of the minimum-link covering trails for the two-dimensional grid G 2 n := {0, 1, . . . , n − 1} × {0, 1, . . . , n − 1}. Furthermore, we introduce a general algorithm that returns a covering cycle of analogous link length for any even value of n. Finally, we provide the tight upper bound n 2 − 3 + 5 • √ 2 units for the minimum total distance travelled to visit all the nodes of G 2 n with a minimum-link trail (i.e., a trail with 2 • n − 2 edges if n is above two).