Adjacency-Preserving 2D Grid Embedding
Embedding planar graphs (with max degree four) in an adjacency-preserving way onto a grid is NP-complete, meaning that there's unlikely to be simple necessary and sufficient conditions. Actually that's still true even for embedding trees into a grid. See:

S. Bhatt and S. Cosmodakis. The complexity of minimizing wire lengths in VLSI layouts.
Inform. Proc. Lett. 25:263–267, 1987.