By density functional theory (DFT) calculations, the early stages of the growth of graphene on copper (1 1 1) surface are investigated. At the very first time of graphene growth, the carbon atom sinks into subsurface. As more carbon atoms are adsorbed nearby the site, the sunken carbon atom will spontaneously form a dimer with one of the newly adsorbed carbon atoms, and the formed dimer will up-float on the top of the surface. We emphasize the role of the co-operative relaxation of the co-adsorbed carbon atoms in facilitating the sinking and up-floating of carbon atoms. In detail: when two carbon atoms are co-adsorbed, their co-operative relaxation will result in different carbon-copper interactions for the co-adsorbed carbon atoms. This difference facilitates the sinking of a single carbon atom into the subsurface. As a third carbon atom is co-adsorbed nearby, it draws the sunken carbon atom on top of the surface, forming a dimer. Co-operative relaxations of the surface involving all adsorbed carbon atoms and their copper neighbors facilitate these sinking and up-floating processes. This investigation is helpful for the deeper understanding of graphene synthesis and the choosing of optimal carbon sources or process. (C) 2013 Elsevier B. V. All rights reserved.