If G is a solvable group, we take Δ(G) to be the character degree graph for G with primes as vertices. We prove that if Δ(G) is a square, then G must be a direct product.
@article{arxiv.1105.2799,
title = {Square character degree graphs yield direct products},
author = {Mark L. Lewis and Qingyun Meng},
journal= {arXiv preprint arXiv:1105.2799},
year = {2014}
}