Reaction pathways are presented for hydrogen-mediated isomerization of a zigzag graphene edge containing a five-member carbon ring surrounded by six-member rings. A new reaction sequence in which this embedded five-member ring moves, or migrates, through the edge has been identified. The elementary steps of the pathways were analyzed using density functional theory (DFT). Rate coefficients were obtained by classical transition state theory utilizing the DFT energies, frequencies, and geometries. The results indicate that this new reaction sequence is competitive with the other important zigzag edge reactions allowing embedded five-member rings to move freely within a zigzag edge. The embedded rings have slight thermodynamic preference for the interior of the edge over the corner for large substrates.