Abstract Opinion diffusion is studied on social graphs where agents hold binary opinions and where social pressure leads them to conform to the opinion manifested by the majority of their neighbors. Within this setting, questions related to whether a minority/majority can spread the opinion it supports to all the other agents are considered. It is shown that, no matter of the underlying graph, there is always a group formed by a half of the agents that can annihilate the opposite opinion. Instead, the influence power of minorities depends on certain features of the given graph, which are NP-hard to be identified. Deciding whether the two opinions can coexist in some stable configuration is NP-hard, too.