Computational methods to engineer cellular metabolism promise to play a critical role in producing pharmaceutical, repairing defective genes, destroying cancer cells, and generating biofuels. Elementary Flux Mode (EFM) analysis is one such powerful technique that has elucidated cell growth and regulation, predicted product yield, and analyzed network robustness. EFM analysis, however, is a computationally daunting task because it requires the enumeration of all independent and stoichiometrically balanced pathways within a cellular network. We present in this paper an EFM enumeration algorithm, termed graphical EFM or gEFM. The algorithm is based on graph traversal, an approach previously assumed unsuitable for enumerating EFMs. The approach is derived from a pathway synthesis method proposed by Mavrovouniotis et al. The algorithm is described and proved correct. We apply gEFM to several networks and report runtimes in comparison with other EFM computation tools. We show how gEFM benefits from network compression. Like other EFM computational techniques, gEFM is sensitive to constraint ordering; however, we are able to demonstrate that knowledge of the underlying network structure leads to better constraint ordering. gEFM is shown to be competitive with state-of-the-art EFM computational techniques for several networks, but less so for networks with a larger number of EFMs.