Metzger2025-09-082025-09-082025-09-03https://hdl.handle.net/1773/53829Minimal-genus graph embedding is about drawing graphs on surfaces with no edges crossing and as few holes as possible. This thesis first covers the necessary background in topological graph theory to understand graph embeddings through rotation systems. It then studies an adaptation of this approach, Practical Algorithm for Graph Embedding (PAGE), that takes advantage of the cycle sequence of a graph to work more efficiently in practice, especially for graphs of high girth or low degree. This enables it to determine the previously intractable genus of the (3, 12)-cage as 17.Attribution 3.0 United Stateshttp://creativecommons.org/licenses/by/3.0/us/Thesis