Complexity dichotomy on partial grid recognition
Abstract
Deciding whether a graph can be embedded in a grid using only unit-length 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.
Cite
@article{arxiv.1006.3541,
title = {Complexity dichotomy on partial grid recognition},
author = {Vinícius G. P. de Sá and Guilherme D. da Fonseca and Raphael Machado and Celina M. H. de Figueiredo},
journal= {arXiv preprint arXiv:1006.3541},
year = {2012}
}