Elementary Flux Mode (EFM) analysis is a fundamental network decomposition technique used for cellular pathway analysis in Systems Biology and Metabolic Engineering. EFM analysis has been utilized to examine robustness, regulation and microbial stress responses, to increase product yield, and to assess plant FItness and agricultural productivity. An EFM is a thermodynamically feasible path operating at steady state in a biochemical network, and is independent of other EFMs in the sense that it cannot be generated as a non-negative linear combination of other EFMs. We present in this paper a pathway analysis algorithm, termed graphical EFM or gEFM, based on graph traversal. Graph theoretical approaches were previously assumed to be less competitive than techniques based on the double-description method, a computational technique used for enumerating the extreme rays of a pointed cone. Importantly, we show that a practical graph-based traversal approach for computing EFMs is competitive with existing techniques. Applied to several biochemical networks, we show runtime speedups in the range of 2.5× to 31× when compared to the state-of-the-art tool (EFMTool).