Complexity dichotomy on partial grid recognition

Deciding whether a graph can be embedded in a grid using only unitlength edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even trivial. A natural step, outstanding thus far, was to provide a broad classification of graphs that make for polynomial or NP-complete instances. We provide such a classification based on the set of allowed vertex degrees in the input graphs, yielding a full dichotomy on the complexity of the problem. As byproducts, the previous NP-completeness result for binary trees was strengthened to strictly binary trees, and the three-dimensional version of the problem was for the first time proven to be NP-complete. Our results were made possible by introducing the concepts of consistent orientations and robust gadgets, and by showing how the former allows NP-completeness proofs by local replacement even in the absence of the latter.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Vinícius G. P. de Sá, Guilherme Dias da Fonseca, Raphael Machado, Celina M. Herrera de Figueiredo
• Keywords: Binary Trees, CORR 2010, Education, NP-complete, NP-complete Instances