Density functional calculations are performed to study the H-atom diffusion on titanium dioxide (110) surface in the cases of water-molecule dissociation and splitting of the adjacent hydroxyl OH pair. It is shown that, when a water molecule is adsorbed at a surface oxygen-vacancy site, a fragment H atom of the water molecule tends to diffuse toward the nearest-neighboring bridging-oxygen sites by using a straight-line or relay-point path. As the result, a pair of surface hydroxyl OH is formed on the same oxygen row. In a thermal process, on the other hand, such OH pair favorably splits only by using a relay-point path, i.e., by transferring one H atom from a bridging-oxygen site to a next-neighboring one along the same oxygen row by way of another in-plane oxygen site. We found that the latter splitting reaction is activated around room temperature.