In a social network, countering false news or rumor is one of the biggest challenges. In this study, the rumor diffusion in the homogeneous and heterogeneous social networks has been explored, introducing the concept of an anti-rumor spreading agent as the mitigator. The study consists of four types of individuals within the population, namely, spreaders, mitigators, ignorants, and stiflers. To model the interactions among the aforesaid populations, five mathematical models have been proposed for the rumor diffusion process in both homogeneous and heterogeneous environments. The proposed SMIR model has been solved using Runge-Kutta fourth-order method and then the model has been extended into a fuzzy environment to deal with the inherent uncertainties. The spreading of a rumor is a socio-psychological phenomenon which may contain delay in the process. To investigate this, the SMIR model has been transformed into a delayed SMIR model. In the heterogeneous environment, the SMIR model has been extended in a small world and scale-free networks. The results obtained are analyzed in all the cases and found that mitigator plays a critical role to control the spread of rumor in homogeneous and heterogeneous environments. •Objective: to explore the spread of rumors on social networks.•Four population: spreaders, mitigators, ignorants, and stiflers.•The fuzzy paradigm has been used to remove uncertainties.•The results are obtained from homogeneous and heterogeneous environmnet.