By utilizing the reduced contact area of nanowires, we show that epitaxial growth of a broad range of semiconductors on graphene can in principle be achieved. A generic atomic model is presented which describes the epitaxial growth configurations applicable to all conventional semiconductor materials. The model is experimentally verified by demonstrating the growth of vertically aligned GaAs nanowires on graphite and few-layer graphene by the self-catalyzed vapor-liquid-solid technique using molecular beam epitaxy. A two-temperature growth strategy was used to increase the nanowire density. Due to the self-catalyzed growth technique used, the nanowires were found to have a regular hexagonal cross-sectional shape, and are uniform in length and diameter. Electron microscopy studies reveal an epitaxial relationship of the grown nanowires with the underlying graphitic substrates. Two relative orientations of the nanowire side-facets were observed, which is well explained by the proposed atomic model. A prototype of a single GaAs nanowire photodetector demonstrates a high-quality material. With GaAs being a model system, as well as a very useful material for various optoelectronic applications, we anticipate this particular GaAs nanowire/graphene hybrid to be promising for flexible and low-cost solar cells.