While graphene has attracted significant attention from the research community due to its high charge carrier mobility, important issues remain unresolved that prevent its widespread use in technologically significant applications such as digital electronics. For example, the chemical inertness of graphene hinders integration with other materials, and the lack of a bandgap implies poor switching characteristics in transistors. The formation of ordered organic monolayers on graphene has the potential to address each of these challenges. In particular, functional groups incorporated into the constituent molecules enable tailored chemical reactivity, while molecular-scale ordering within the monolayer provides sub-2 nm templates with the potential to tune the electronic band structure of graphene via quantum confinement effects. Toward these ends, we report here the formation of well-defined one-dimensional organic nanostructures on epitaxial graphene via the self-assembly of 10,12-pentacosadiynoic acid (PCDA) in ultrahigh vacuum (UHV). Molecular resolution UHV scanning tunneling microscopy (STM) images confirm the one-dimensional ordering of the as-deposited PCDA monolayer and show domain boundaries with symmetry consistent with the underlying graphene lattice. In an effort to further stabilize the monolayer, in situ ultraviolet photopolymerization induces covalent bonding between neighboring PCDA molecules in a manner that maintains one-dimensional ordering as verified by UHV STM and ambient atomic force microscopy (AFM). Further quantitative insights into these experimental observations are provided by semiempirical quantum chemistry calculations that compare the molecular structure before and after photopolymerization.