Hydrogen abstraction reactions by methyl radicals on the zigzag and armchair edges of perylene are studied by density functional theory (DFT) to explore various growth pathways that seem to be in line with experimental observations. The DFT approach is validated by comparing the results obtained from calculations with six different functionals with those obtained from correlated ab initio methods, thereby emphasizing the calculation of reaction barriers. A useful compromise between accuracy and computational cost is provided by DFT, and possible pathways are studied in detail at this level of calculation. Our computational study is carried out by ordering, as a first step, all of the isomers that arise from the abstraction of one or two H atoms from 1,12-dimethyl-1,12-dihydroperylene and 3,4-dimethyl-3,4-dihydroperylene with respect to their energies. Subsequently, only those pathways that connect low-energy isomers are investigated. The calculations reveal that the selected pathways are favored thermodynamically, and also that the reaction barriers are somewhat higher than the energy locally available for the respective reaction. Notably, in the case of 3,4-dimethyl-3,4-dihydroperylene, the first two reaction steps have no or only a very low reaction barrier. The final conclusion of our study is that a cascade of reactions is possible that leads to the growth of a graphene sheet on a graphite surface.