Edge flames obtained on a hydrogen/air non-premixed opposed-jet burner after the local extinction of the disk-shaped diffusion flame are investigated with 2-D direct numerical simulations using detailed chemical kinetics and transport. Over a large range of flowrates, edge flames were found to coexist with the well-known strongly burning diffusion flames corresponding to the upper branch of the S-shaped curve. The critical flowrates of the strong hysteresis associated with the transitions between the two solution branches were identified: re-establishment of the diffusion flame is controlled by the propagation of the edge flame and cannot be represented simply by the extinction scalar dissipation rate. It was also observed that in all the flow conditions simulated, the edge flame was able to consume all the supplied fuel by re-orienting itself, varying its flame surface area, or changing its structure. The latter was found to depend on the flow conditions (which strongly affects the degree of mixing ahead of the edge flame) and can take on different configurations ranging from a triple flame to an essentially premixed flame. Because of flame curvature and the preferential diffusion of hydrogen, the propagation speed of the edge flames was found to be higher than that of the corresponding planar premixed flames.