NP-completeness of determining unit distance graphs with integer coordinates

TL;DR

This work proves that deciding whether a graph can be realized as a unit-distance graph with integer coordinates is NP-complete. It achieves this by adapting the Eades–Whitesides logic-engine gadget to a square lattice, encoding NA3SAT as a frame–axis–armature construction whose realization in $\mathbb{Z}^2$ corresponds to a satisfying assignment. The reduction yields a polynomial-time certificate of embeddability and demonstrates NP-hardness by a direct correspondence between formula satisfiability and a non-overlapping lattice embedding. The result clarifies the complexity border for restricted unit-distance realizations and suggests avenues for extending the approach to other lattices or precision-based variants.

Abstract

The problem of determining whether a graph G can be realized as a unit-distance graph in $\mathbb{R}^2$ with integer-valued coordinates is NP-complete. We implement Eades and Whitesides' logic engine in this setting, and construct a graph that is realizable if and only if an arbitrary NA3SAT formula is satisfiable.

Figures (5)

• Figure 1: An example logic engine showing the NA3SAT formula $(X_1, X_2, X_3), (X_1, X_2, X_4), (\neg X_1, X_3, X_4)$ satisfied by $X_1 = X_3 = 0, X_2 = X_4 = 1$
• Figure 2: The frame and horizontal axis of the logic engine
• Figure 3: How the chains are placed around the axis
• Figure 4: Different possible realizations of a pair of flags on adjacent chains
• Figure 5: The graph $G$ corresponding to the logic engine in figure \ref{['fig:logic_engine_example']}