Zig-Zag Numberlink is NP-Complete

When can t terminal pairs in an m× n grid be connected by t vertex-disjoint paths that cover all vertices of the grid? We prove that this problem is NP-complete. Our hardness result can be compared to two previous NP-hardness proofs: Lynch’s 1975 proof without the “cover all vertices” constraint, and Kotsuma and Takenaga’s 2010 proof when the paths are restricted to have the fewest possible corners within their homotopy class. The latter restriction is a common form of the famous Nikoli puzzle Numberlink ; our problem is another common form of Numberlink, sometimes called Zig-Zag Numberlink and popularized by the smartphone app Flow Free.