Phylogenetic networks play an increasingly important role in the comparative analysis of many systems. These include studies of hybridization, linguistic exchange, bacterial and viral evolution, and the origins of zoonotic disease. Computationally, these problems present many NP-hard optimizations requiring heuristic procedures to generate useful results. Here, algorithmic approaches to generate heuristic solutions are discussed and recommendations for real-world use cases are presented for the phylogenetic graph search tool Phylogenetic Graph (PhyG).