Abstract. A graph is radially maximal if its radius decreases after the addition of any edge of its complement. A superclass of radially maximal graphs is a class of two-radially maximal graphs. A graph is two-radially maximal if it is noncomplete and for each pair u,v of its vertices at distance two the addition of the new edge uv decreases its radius. We prove that the central subgraph of any two-radially maximal graph contains an edge. Moreover, those of them that have a star as the central subgraph are sequential joins of complete graphs.