In this paper, we study an information exchange process in which a network of individuals exchanges a binary opinion. In the process, the individuals change their opinions only if a majority of their neighbours have the opposite opinion and they do it synchronously. Motivated by applications in multiagent systems, distributed computing, and social science, our goal is to derive graph-theoretic features of the network that guarantee whenever a majority of individuals initially have the same opinion, they will eventually spread the opinion to all individuals. We tackle the problem by first introducing a graph-theoretic notion called controlling set which is capable of characterising the information exchange process and, by exploiting the notion, we obtain a series of lower and upper bounds on the in-degree of vertices as well as lower bound on the size of certain neighbourhoods for guaranteeing the majority to unanimity behaviour.