Complexity dichotomy on partial grid recognition

TL;DR

This paper classifies the computational complexity of partial grid recognition based on vertex degrees, establishing a complete dichotomy between polynomial and NP-complete cases, and extends known results to binary and 3D cases.

Contribution

It provides a full complexity classification for partial grid recognition based on vertex degrees, introduces concepts of consistent orientations and robust gadgets, and strengthens previous NP-completeness results.

Findings

Full dichotomy classification based on vertex degrees

NP-completeness for binary and 3D cases established

Introduction of consistent orientations and robust gadgets

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.