The high affinity of graphene for H₂S makes it a potential adsorbent for separating H₂S from industrial waste gas streams, and for its use in H₂S sensors. The gas streams containing H₂S also contain S atoms that can get bonded to graphene during adsorption. Moreover, at high temperatures, H₂S dissociate at graphene defects leading to H₂ desorption with S atom remaining on graphene. This study reports the effect of the presence of S atom on graphene on H₂S adsorption using plane-wave density functional theory to assess the suitability of graphene-based adsorbents for continuous H₂S capture. During H₂S interaction with pristine graphene with S atom bonded to it, a surface cleaning mechanism was observed, where H₂S exothermically removed S atom by forming H₂S₂. When S atom was doped in the single-vacancy defect of graphene, both physisorption and chemisorption of H₂S was observed. The adsorption energy released during H₂S physisorption on S-doped graphene was comparable to that for pristine graphene. However, during chemisorption, H₂S dissociated on S-doped graphene to form HSSH bonded to graphene with the release of a high amount of energy. The results suggest that S atom in single-vacancy defects can enhance H₂S capture on graphene.