Zig-Zag Numberlink is NP-Complete
Aaron Adcock, Erik D. Demaine, Martin L. Demaine, Michael P. O'Brien,, Felix Reidl, Fernando S\'anchez Villaamil, and Blair D. Sullivan

TL;DR
This paper proves that the Zig-Zag Numberlink puzzle, involving connecting terminal pairs with disjoint paths covering all grid vertices, is NP-complete, establishing its computational difficulty.
Contribution
It demonstrates NP-completeness of Zig-Zag Numberlink, a variant of Numberlink, expanding understanding of its computational complexity.
Findings
Proves NP-completeness of Zig-Zag Numberlink.
Connects the problem to previous NP-hardness results.
Highlights the problem's relation to popular puzzles and applications.
Abstract
When can terminal pairs in an grid be connected by 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 \emph{Numberlink}; our problem is another common form of Numberlink, sometimes called \emph{Zig-Zag Numberlink} and popularized by the smartphone app \emph{Flow Free}.
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Taxonomy
TopicsComplexity and Algorithms in Graphs
