Cascading phenomena have been studied extensively in various networks. Particularly, it has been shown that the community structures in networks impact their cascade processes. However, the role of community structures in cascading failures in power grids have not been studied heretofore. In this paper, cascading failures in power grids are studied using interaction graphs. Key evidence has been provided that the community structures in interaction graphs bear critical information about the cascade process and the role of system components in cascading failures in power grids. Further, a centrality measure based on the community structures is proposed to identify critical components of the system, which their protection can help in containing failures within a community and prevent the propagation of failures to large sections of the power grid. Various criticality evaluation techniques, including data-driven, epidemic simulation-based, power system simulation-based and graph-based, have been used to verify the importance of the identified critical components in the cascade process and compare them with those identified by traditional centrality measures. Moreover, it has been shown that the loading level of the power grid impacts the interaction graph and consequently, the community structure and criticality of the components in the cascade process.