Any graph with 9 or fewer vertices that can be embedded on a torus and linklessly in 3D space can also be embedded linklessly within the standard unknotted torus.
This paper presents a construction method for embedding complete graphs on surfaces where the dual graph has low connectivity, specifically a cutvertex, and the genus of the embedding is close to the theoretical minimum.
Graph Embeddings zur Bewältigung von Sampling-Bias in ökologischen Netzwerken.